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Problem 201

$\left.\begin{array}{l}\text { Suppose we're finding the steady state vector for the transition matrix } A=\left[\begin{array}{ll}0.93 & 0.05 \\ 0.07 & 0.95\end{array}\right] \text {, and upon performing some row operations on the transition } \\ \qquad-0.07 \\ -0.05\end{array}\right]$ matrix we obtain: $\left[\begin{array}{cc}-0.07 & 0.05 \\ 0 & 0\end{array}\right]$ . What is the actual steady state vector? [This question is based on your assigned pre-reading/prep for the upcoming Assignment] None of these $\left[\begin{array}{l}1 \\ \frac{5}{7}\end{array}\right]$ $\left[\begin{array}{l}\frac{5}{7} \\ 1\end{array}\right]$ $\square\left[\begin{array}{c}\frac{7}{12} \\ \frac{5}{12}\end{array}\right]$ $\square\left[\begin{array}{c}\frac{5}{12} \\ \frac{7}{12}\end{array}\right]$

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Problem 202

Which of the following are TRUE? [This question is based on your assigned pre-reading/prep for the upcoming Assignment] Transition matrices must be square. Entries in rows in a transition matrix add to 1. $\left[\begin{array}{ll}0.7 & 0.25 \\ 0.3 & 0.75\end{array}\right]$ could be a transition matrix. $\left[\begin{array}{cc}1.1 & -0.4 \\ -0.1 & 1.4\end{array}\right]$ could be a transition matrix.

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Problem 203

(a) Find the eigenvalues of for
Eigenvalues of $A=\left[\begin{array}{rrrr} 2 & 1 & 4 & 3 \\ 0 & 14 & 3 & 8 \\ 0 & 0 & 18 & 9 \\ 0 & 0 & 0 & 7 \end{array}\right]$ are: $\square$ 졸 $\square$ 줘몯 $\square$ $\square$ (in increasing order) The entry boxes with a small icon beside them are designed to accept numbers or formulas. Help | Switch to Equation Editor

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Problem 204

Given a $2 \times 2$ matrix that has the eigenvalues 2 and -7 , and the eigenvectors $\left[\begin{array}{c}-1 \\ 3\end{array}\right]$ and $\left[\begin{array}{c}-9 \\ -8\end{array}\right]$ respectively, which of the following could represent $P$ and $D$ ? $P=\left[\begin{array}{cc}-1 & -9 \\ 3 & -8\end{array}\right]$ and $D=\left[\begin{array}{cc}2 & 0 \\ 0 & -7\end{array}\right]$ $P=\left[\begin{array}{cc}-1 & 3 \\ -9 & -8\end{array}\right]$ and $D=\left[\begin{array}{cc}2 & 0 \\ 0 & -7\end{array}\right]$ $P=\left[\begin{array}{cc}-1 & -9 \\ 3 & -8\end{array}\right]$ and $D=\left[\begin{array}{cc}-7 & 0 \\ 0 & 2\end{array}\right]$ $P=\left[\begin{array}{cc}-9 & -1 \\ -8 & 3\end{array}\right]$ and $D=\left[\begin{array}{cc}2 & 0 \\ 0 & -7\end{array}\right]$

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Problem 205

If $C=\left[\begin{array}{ll}4 & 2\end{array}\right], D=\left[\begin{array}{ll}-7 & -3\end{array}\right]$ , and $E=\left[\begin{array}{lll}1 & 2 & -3 \\ 5 & 6 & -9\end{array}\right]$ , is the following statement true or false? $(C+D) E=C E+D E$ true false

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Problem 206

11. Galoulate the determinant. $\left[\begin{array}{ll} 5 & 2 \\ 3 & 1 \end{array}\right]$ 1 11 0 $-11$ $-1$

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Problem 207

Multiply the matrices $S=\begin{pmatrix} 4 & 1 \\ -1 & -2 \end{pmatrix}$ and $T=\begin{pmatrix} -5 & 0 \\ -4 & 3 \end{pmatrix}$ to show $S T \neq T S$ . Fill in the boxes: $S T = \_\_\_\_ \quad T S = \_\_\_\_$ .

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Problem 208

Calculate the products of matrices $S$ and $T$ : $S T$ and $T S$ to show multiplication is not commutative. Fill in the boxes.

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Problem 209

Find the product matrix $P Q$ for the matrices $P=\begin{bmatrix} 2 & 7 \\ 1 & 7 \\ 0 & 4 \\ -1 & -3 \end{bmatrix}$ and $Q=\begin{bmatrix} -2 \\ -3 \end{bmatrix}$ .

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Problem 210

Find the matrix result of $m \times H$ given the equations involving matrices and constants.

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Problem 211

Find the product matrix $B C$ for matrices $B=\begin{pmatrix}-2 & 3 \\ 0 & 5\end{pmatrix}$ and $C=\begin{pmatrix}0 & 4 & -1 \\ 3 & 0 & 1\end{pmatrix}$ .

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Problem 212

Show that matrix multiplication is not commutative by calculating $S T$ and $T S$ for the matrices $S=\begin{bmatrix} 4 & 1 \\ -3 & -2 \end{bmatrix}$ and $T=\begin{bmatrix} 0 & 4 \\ -4 & 3 \end{bmatrix}$ .

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Problem 213

Find the dimensions of the product matrix formed by multiplying any two of the following matrices: $A$ , $B$ , $C$ , $D$ .

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Problem 214

Find the matrix resulting from $m \times H$ given the equations involving matrices.

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Problem 215

Find the general solution for the system with the matrix:
$\left[\begin{array}{rrrr} 1 & 4 & 1 & 2 \\ 4 & 16 & -1 & -17 \end{array}\right]$
Choose A, B, C, or D for the solution type.

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Problem 216

Find the values of $h$ for which the matrix is consistent: $\begin{bmatrix} 1 & h & 4 \\ -3 & 6 & -16 \end{bmatrix}$ Choose A, B, C, or D.

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Problem 217

Row reduce the matrix and find pivot positions. Given matrix: $\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \end{array}\right]$

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Problem 218

Row reduce the matrix below to reduced echelon form and identify pivot positions.
$\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \end{array}\right]$

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Problem 219

Row reduce the matrix and identify pivot positions. Which option shows the correct reduced echelon form?
$\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \end{array}\right]$
A. $\left[\begin{array}{rrrr}1 & 0 & -1 & -2 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0\end{array}\right]$
B. $\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1\end{array}\right]$
C. $\left[\begin{array}{llll}1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 5 \\ 0 & 0 & 0 & 0\end{array}\right]$
D. $\left[\begin{array}{llll}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & 6\end{array}\right]$

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Problem 220

Find values of h for which the matrix is consistent: $\left[\begin{array}{rrr} 1 & 2 & -4 \\ 5 & h & -20 \end{array}\right]$ Choose A, B, C, or D.

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Problem 221

Find the general solution for the system with the augmented matrix:
$\left[\begin{array}{llll} 1 & 2 & 0 & 3 \\ 3 & 5 & 0 & 5 \end{array}\right]$
Choose the correct option for $x_1$ , $x_2$ , and $x_3$ .

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Problem 222

Identify if the following matrices are in reduced echelon form, echelon form, or neither: a. $\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 5 & 0 & 0 \\ 0 & 0 & 1 & 1\end{bmatrix}$ b. $\begin{bmatrix}1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0\end{bmatrix}$ c. $\begin{bmatrix}1 & 3 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$

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Problem 223

Solve the system from the matrix: $\begin{bmatrix} 0 & 1 & -3 & 6 \\ 1 & -3 & 7 & -14 \end{bmatrix}$ Choose the correct solution set or state if there's no solution.

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Problem 224

Find values of $h$ for which the matrix is consistent: $\left[\begin{array}{rrr} 1 & 4 & -3 \\ 2 & h & -6 \end{array}\right]$ Choices: A. $h \neq$ B. $h=$ C. for all $h$ D. for no $h$ .

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Problem 225

Identify if the following matrices are in reduced echelon form or just echelon form: a. $\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 1 & 1\end{bmatrix}$ b. $\begin{bmatrix}1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$ c. $\begin{bmatrix}1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$ Classify matrix a.

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Problem 226

Find the general solution for the system with the matrix:
$\left[\begin{array}{rrrr} 0 & 1 & -2 & 2 \\ 1 & -4 & 6 & -5 \end{array}\right]$
Choose A, B, C, or D for your answer.

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Problem 227

Find the general solution for the system with the augmented matrix:
$\left[\begin{array}{llll} 1 & 1 & 0 & -3 \\ 2 & 1 & 0 & -9 \end{array}\right]$
Choose A, B, C, or D based on the solution.

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Problem 228

Solve the system by using the inverse of the coefficient matrix. $\begin{array}{l} -5 x-y+2 z=4 \\ 4 x+9 y-5 z=-12 \\ 2 x-5 y+z=2 \end{array}$

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Problem 229

11.2 Exercises Answers to selected odd-numbered problems begin on page ANS-28.
In Problems 1-8, the general solution of the linear system
1. $\mathbf{A}=\left(\begin{array}{ll}-2 & -2 \\ -2 & -5\end{array}\right), \quad \mathbf{X}(t)=c_{1}\binom{2}{-1} e^{-t}+c_{2}\binom{1}{2} e^{-6 t}$ $\mathbf{X}^{\prime}=\mathbf{A X}$ is given. (a) In each case discuss the nature of the solution in a
2. $\mathbf{A}=\left(\begin{array}{rr}-1 & -2 \\ 3 & 4\end{array}\right), \quad \mathbf{X}(t)=c_{1}\binom{1}{-1} e^{t}+c_{2}\binom{-4}{6} e^{2 t}$ neighborhood of $(0,0)$ . (b) With the aid of a graphing utility plot the solution that
3. $\mathbf{A}=\left(\begin{array}{rr}1 & -1 \\ 1 & 1\end{array}\right), \quad \mathbf{X}(t)=e^{t}\left[c_{1}\binom{-\sin t}{\cos t}+c_{2}\binom{\cos t}{\sin t}\right]$ satisfies $\mathbf{X}(0)=(1,1)$ .

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Problem 230

Решить матричное уравнение $\mathrm{XB}-2 \mathrm{X}=\mathrm{A}$ , где матрицы A и B заданы: $A=\left(\begin{array}{ccc} 2 & 3 & -1 \\ 4 & -2 & 5 \\ 0 & 7 & 3 \end{array}\right), \quad B=\left(\begin{array}{ccc} 6 & -3 & 2 \\ 0 & 1 & 8 \\ -4 & 7 & 3 \end{array}\right)$
В ходе решения при помощи матричных операций получить и обосновать аналитическую формулу для неизвестной матрицы X и только после этого найти числовые значения для элементов этой матрицы.

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Problem 231

Q3: Find the Rank. 1. $\left[\begin{array}{rrr} 6 & -4 & 0 \\ -4 & 0 & 2 \\ 0 & 2 & 6 \end{array}\right]$ 2. $\left[\begin{array}{rrrr} 2 & 4 & 8 & 16 \\ 16 & 8 & 4 & 2 \\ 4 & 8 & 16 & 2 \\ 2 & 16 & 8 & 4 \end{array}\right]$
Q4: Are the following sets of vectors linearly independent? Show the details of your work.
1. $\left[\begin{array}{lll}0 & 1 & 1\end{array}\right],\left[\begin{array}{lll}1 & 1 & 1\end{array}\right],\left[\begin{array}{lll}0 & 0 & 1\end{array}\right]$
2. $\left[\begin{array}{lll}4 & -1 & 3\end{array}\right],\left[\begin{array}{lll}0 & 8 & 1\end{array}\right],\left[\begin{array}{lll}1 & 3 & -5\end{array}\right],\left[\begin{array}{lll}2 & 6 & 1\end{array}\right]$
Q5: Showing the details, evaluate: 1. $\left|\begin{array}{rrrr} 4 & 7 & 0 & 0 \\ 2 & 8 & 0 & 0 \\ 0 & 0 & 1 & 5 \\ 0 & 0 & -2 & 2 \end{array}\right|$ 2. $\left|\begin{array}{rrr} 4 & -1 & 8 \\ 0 & 2 & 3 \\ 0 & 0 & 5 \end{array}\right|$

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Problem 232

Za koji realan parametar $b$ dani sustav ima jedinstveno rješenje? $\begin{array}{r} x_{1}+2 x_{2}+(b+1) x_{3}=1 \\ -2 x_{1}+b x_{2}-4 x_{3}=0 \\ x_{1}+x_{2}+2 x_{3}=0 \end{array}$
Odaberite jedan odgovor: a. $\quad b \neq 1$ i $b \neq-1$ b. $\quad b \neq 2$ i $b \neq-1$ c. $b \neq 2$ i $b \neq 1$ d. Ne znam. e. $\quad b \neq-2$ i $b \neq 1$

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Problem 233

Find the general solution of the system $\mathbf{x}^{\prime}=\left(\begin{array}{rr} -2 & 1 \\ 1 & -2 \end{array}\right) \mathbf{x}+\binom{2 e^{-t}}{3 t}=\mathbf{A} \mathbf{x}+\mathbf{g}(t)$

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Problem 234

Find matrix $C$ such that $A B + C^{T} = \left(\begin{array}{ccc}-10 & -17 & -8 \\ 10 & 20 & 15 \\ 0 & -9 & -6\end{array}\right)$ , where $A = \left(\begin{array}{cc}1 & -3 \\ 0 & 5 \\ 2 & -1\end{array}\right)$ and $B = \left(\begin{array}{ccc}1 & -2 & 1 \\ 2 & 5 & 3\end{array}\right)$ .

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Problem 235

$\checkmark 1$ 2 3 4 $\checkmark 5$ 6
Let $A=\left[\begin{array}{cc}1 & 2 \\ -2 & -3\end{array}\right]$ and $B=\left[\begin{array}{cc}3 & 0 \\ -1 & -1\end{array}\right]$ . Find each matrix below. If a matrix is not defined, click on "Undefined".

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Problem 236

Given a $2 \times 2$ matrix that has the eigenvalues -3 and -5 , and the eigenvectors $\left[\begin{array}{l}6 \\ 3\end{array}\right]$ and $\left[\begin{array}{l}-7 \\ -8\end{array}\right]$ respectively, which of the following could represent $P$ and $D$ ? $P=\left[\begin{array}{ll}6 & -7 \\ 3 & -8\end{array}\right]$ and $D=\left[\begin{array}{cc}-3 & 0 \\ 0 & -5\end{array}\right]$ $P=\left[\begin{array}{cc}6 & 3 \\ -7 & -8\end{array}\right]$ and $D=\left[\begin{array}{cc}-3 & 0 \\ 0 & -5\end{array}\right]$ $P=\left[\begin{array}{ll}-7 & 6 \\ -8 & 3\end{array}\right]$ and $D=\left[\begin{array}{cc}-3 & 0 \\ 0 & -5\end{array}\right]$ $P=\left[\begin{array}{ll}6 & -7 \\ 3 & -8\end{array}\right]$ and $D=\left[\begin{array}{cc}-5 & 0 \\ 0 & -3\end{array}\right]$

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Problem 237

Determine the rank of the matrix $\left[\begin{array}{ccc}1 & -3 & 5 \\ -3 & 9 & -15 \\ 2 & -6 & 10\end{array}\right]$ 3
0
4 2

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Problem 238

Let $T: R^{2} \rightarrow R^{2}$ be the linear operator defined by $T\left(\left[\begin{array}{c} x_{1} \\ x_{2} \end{array}\right]\right)=\left[\begin{array}{c} 5 x_{1}-x_{2} \\ 5 x_{1}+x_{2} \end{array}\right]$ and let $B=\left\{\boldsymbol{u}_{1}, \boldsymbol{u}_{2}\right\}$ be the basis for which $\boldsymbol{u}_{1}=\left[\begin{array}{l}1 \\ 5\end{array}\right], \boldsymbol{u}_{2}=\left[\begin{array}{c}-1 \\ 0\end{array}\right]$ . Find $[T]_{B}$ . $[T]_{B}=(\square$

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Problem 239

If $A$ is a $3 \times 3$ matrix such that $\operatorname{det}(A)=7$ , then: $\operatorname{det}\left(2 A^{T} A^{-1}\right)=$
Select one: a. 56 b. 8 c. 98 d. 14

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Problem 240

Evaluate the determinant of the matrix. $E=\left[\begin{array}{ll} 4 & 0 \\ 7 & 0 \end{array}\right]$
The determinant is $\square$ .

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Problem 241

If $A$ is a $3 \times 3$ matrix such that $\operatorname{det}(A)=3$ then $\operatorname{det}\left(A^{2}\right)=6$ .
Select one: True False

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Problem 242

If the adjoint of amatrix is $\left(\begin{array}{cc}2 & -3 \\ 4 & 5\end{array}\right)$ find the matrix.

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Problem 243

MY NOTES LARPCALC11 8.2.031.
If possible, find $A B$ . (If not possible, enter IMPOSSIBLE in any cell of the matrix.) $\begin{array}{c} A=\left[\begin{array}{rr} -1 & 6 \\ -2 & 6 \\ 0 & 3 \end{array}\right], B=\left[\begin{array}{ll} 3 & 4 \\ 0 & 8 \end{array}\right] \\ A B=\left[\begin{array}{c} \square \\ \square \end{array}\right] \Rightarrow \end{array}$ - $\Rightarrow$
State the dimension of the result. (If not possible, enter IMPOSSIBLE in both answer blank $\square$ $\times$ $\square$ Need Help? Read it

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Problem 244

8. FUNDRAISING The cheerleading squad is raising money for new uniforms by selling popcorn balls and calendars. Tanya raised $\$ 70$ by selling 25 popcorn balls and 30 calendars. Nichole raised $\$ 53$ by selling 20 popcorn balls and 22 calendars. What is the cost of one calendar?
A \ $1 B$ \ $1.25$ C \ $1.50 D$ \ $1.75$
9. Solve the following system of equations using an inverse matrix. $\begin{array}{ccc} -4 x-2 y+z=6 & -x-y-2 z=-3 & 2 x+3 y-z=-4 \\ \text { A }(1,0,-2) & \text { B }(-1,0,-2) & \text { C }(-1,0,2) \\ (1,0,2) & \end{array}$ D
10. If $\mathrm{A}=$ , find the determinant of matrix A .
11. What is the determinant of?
A-8 B 8 C 12

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Problem 245

(b) Given that $\mathrm{M}=\left(\begin{array}{ll}1 & 2 \\ 4 & 3\end{array}\right), \mathrm{N}=\left(\begin{array}{ll}\mathrm{m} & \mathrm{x} \\ \mathrm{n} & \mathrm{y}\end{array}\right)$ and $\mathrm{MN}=\left(\begin{array}{ll}2 & 1 \\ 3 & 4\end{array}\right)$ , find the matrix N .

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Problem 246

Let $A=\left[\begin{array}{ccc} -2 & 2 & 0 \\ 2 & -1 & 1 \\ 3 & -2 & 1 \end{array}\right]$ a. A basis for the row space of $A$ is $\{\square\}$ . You should be able to explain and justify your answer. Enter a coordinate vector, such as $\langle 1,2,3>$ , or a comma separated list of coordinate vectors, such as $\langle 1,2,3\rangle,\langle 4,5,6\rangle$ . b. The dimension of the row space of $A$ is $\square$ because (select all correct answers -- there may be more than one correct answer): A. $\operatorname{rref}(A)$ is the identity matrix. B. Two of the three rows in $\operatorname{rref}(A)$ have pivots. C. The basis we found for the row space of $A$ has two vectors. D. Two of the three columns in $\operatorname{rref}(A)$ are free variable columns. E. $\operatorname{rref}(A)$ has a pivot in every row. F. Two of the threê rows in $\operatorname{rref}(A)$ do not have a pivot. c. The row space of $A$ is a subspace of $\square$ because choose $\square$ d. The geometry of the row space of $A$ is choose $\square$

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Problem 247

Suppose that $A$ is an $n \times m$ matrix of rank 4 , the nullity of $A$ is 2 , and the column space of $A$ is a subspace of $\mathbf{R}^{8}$ . Find the dimensions of $A$ . $A$ has $\square$ rows and $\square$ columns.

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Problem 248

The dot product of two vectors in $\mathbb{R}^{3}$ is defined by $\left[\begin{array}{l} a_{1} \\ a_{2} \\ a_{3} \end{array}\right] \cdot\left[\begin{array}{l} b_{1} \\ b_{2} \\ b_{3} \end{array}\right]=a_{1} b_{1}+a_{2} b_{2}+a_{3} b_{3}$
Let $\vec{v}=\left[\begin{array}{c}0 \\ 9 \\ -6\end{array}\right]$ . Find the matrix $A$ of the linear transformation from $\mathbb{R}^{3}$ to $\mathbb{R}$ given by $T(\vec{x})=\vec{v} \cdot \vec{x}$ . $A=$ $\square$ $\square$ $\square$

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Problem 249

Simplify. $-\frac{1}{4}\left[\begin{array}{ll} 12 & -20 \\ 28 & -12 \end{array}\right]$

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Problem 250

$\text { Let } A=\left(\begin{array}{ccc} 1 & -1 & -1 \\ -1 & 0 & 1 \end{array}\right)$
The matrix transformation associated to $A$ is $T_{A}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ defined by the formula $T_{A}\left(\left[\begin{array}{l} x \\ y \\ z \end{array}\right]\right)=\left[\begin{array}{l} \square \\ \square \end{array}\right]$
Select a blank to input an answer CHECK HELP

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Problem 251

Find the values of $h$ for which the matrix is consistent:
$\left[\begin{array}{rrr} 1 & 2 & -3 \\ 5 & h & -15 \end{array}\right]$
Options: A. $h \neq$ B. $h=$ C. Consistent for all $h$ D. Not consistent for any $h$ .

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Problem 252

Identify if the matrices are in reduced echelon form or only echelon form: a. $\begin{bmatrix}1 & 1 & 0 & 1 & 1 \\ 0 & 5 & 0 & 5 & 5 \\ 0 & 0 & 0 & 3 & 3 \\ 0 & 0 & 0 & 0 & 4\end{bmatrix}$ b. $\begin{bmatrix}1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0\end{bmatrix}$ c. $\begin{bmatrix}1 & 5 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$

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Problem 253

Find the general solution for the system with the matrix:
$\left[\begin{array}{rrrr} 1 & 1 & 0 & -7 \\ 2 & 1 & 0 & -16 \end{array}\right]$
Select A, B, C, or D.

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Problem 254

Find the general solution for the system from the augmented matrix: $\begin{bmatrix} 0 & 1 & -2 & 5 \\ 1 & -4 & 4 & -14 \end{bmatrix}$ Choose A, B, C, or D.

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Problem 255

1 Points] DETAILS MY NOTES LARPCALC11 8.1.076.MI. ASK Y
Use matrices to solve the system of linear equations, if possible. Use Gauss-Jordan elimination. (If there is no solution, enter NO SOLUTION. If there are inf solutions, express $x, y$ , and $z$ in terms of the real number $a$ .) $\begin{array}{c} \left\{\begin{array}{cr} 2 x+2 y-z= & 10 \\ x-3 y+z= & -40 \\ -x+y= & 18 \end{array}\right. \\ (x, y, z)=\left(\begin{array}{c} \square \end{array}\right) \end{array}$ Need Help? Read It Watch It Master It

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Problem 256

Find the inverse of the $3 \times 3$ matrix $A$ $A=\left[\begin{array}{lll} 1 & 2 & 1 \\ 9 & 1 & 0 \\ 3 & 0 & 1 \end{array}\right]$

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Problem 257

An economy is based on three sectors, agriculture, manufacturing, and energy. Production of a dollar's worth of agriculture requires inputs of $\$ 0.20$ from agriculture, $\$ 0.40$ from manufacturing, and $\$ 0.20$ from energy. Product a dollar's worth of manufacturing requires inputs of $\$ 0.30$ from agriculture, $\$ 0.20$ from manufacturing, and $\$ 0.20$ energy. Production of a dollar's worth of energy requires inputs of $\$ 0.20$ from agriculture, $\$ 0.30$ from manufacturi and $\$ 0.30$ from energy.
Find the output for each sector that is needed to satisfy a final demand of $\$ 39$ billion for agriculture, $\$ 48$ billion for manufacturing, and $\$ 76$ billion for energy.
The output of the agricultural sector is $\square$ billion dollars. (Round the final answer to three decimal places as needed. Round all intermediate values to six decimal places as needed.) example Get more help - Clear all

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Problem 258

Write an augmented matrix and use elementary row operations in order to solve the following system of equations. Your final matrix should be in reduced row echelon form. In order to get credit you will have to have a correct final answer as accurate steps in each row operation. $\begin{aligned} 11 x+4 y-6 z & =-4 \\ -6 x-2 y+3 z & =1 \\ 3 x+y-z & =2 \end{aligned}$
Write the augmented matrix:

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Problem 259

MYNOTES LARPCALC11
Use matrices to solve the system of linear equations, if possible. Use terms of the real number a.) $\begin{array}{l} \left\{\begin{array}{l} 2 x-y+3 z= 17 \\ 2 y-z=17 \\ 7 x-5 y=13 \end{array}\right. \\ (x, y, z)=(\square) \end{array}$ Need Help? Read It Watch It

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Problem 260

Use matrices to solve the system of equations (if possible). Use $x$ and $y$ in terms of the real number a.) $\left\{\begin{array}{r} x+2 y=0 \\ x+y=7 \\ 3 x-2 y=16 \end{array}\right.$

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Problem 261

the inverse of $A=\left(\begin{array}{cc} 3 & -5 \\ -2 & 3 \end{array}\right)$

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Problem 262

Which of the following matrices are in echelon form? (A) $\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 0\end{array}\right]$ (B) $\left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1\end{array}\right]$ (C) $\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 0 & 1\end{array}\right]$ (D) $\left[\begin{array}{lll}0 & 1 & 1 \\ 0 & 1 & 1\end{array}\right]$ (5) $\left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 1 & 1\end{array}\right]$

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Problem 263

What are the dimensions of this matrix? $\left[\begin{array}{lll} 3 & 6 & 9 \end{array}\right]$ $\square$ by

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Problem 264

وجد معكوس المصفوفة A $A=\left[\begin{array}{lll} 2 & 0 & 3 \\ 3 & 1 & 4 \\ 1 & 2 & 5 \end{array}\right]$

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Problem 265

Question 1 (1 point) (CAS1U2Q1) Find $\operatorname{det}\left[\begin{array}{cccc}1 & 3 & 0 & -1 \\ 2 & -1 & 2 & 1 \\ 3 & 2 & 0 & 1 \\ -1 & 2 & 1 & 1\end{array}\right]$ 0 $-47$ 23 47 $-23$

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Problem 266

4) $-5\left[\begin{array}{cc}-3 & 0 \\ 0 & 5\end{array}\right]$

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Problem 267

Finding the determinant of a $3 \times 3$ matrix
Evaluate the following determinant. $\left|\begin{array}{ccc} 0 & -4 & 6 \\ 3 & 3 & -3 \\ -3 & 3 & 5 \end{array}\right|$

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Problem 268

a. Give the order of the given matrix. b. If $A=\left[a_{i j}\right]$ , identify $a_{31}$ and $a_{13}$ , if possible. $\left[\begin{array}{rrr} 4 & -8 & 2 \\ -4 & 7 & -3 \end{array}\right]$ a. The order of the given matrix, in the form $m \times n$ , is $\square$ $\square$ .

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Problem 269

I.- Evaluar el siguiente determinante: a) Utilizando la primera fila. b) A partir de la tercera columna. c) Haciendo $a_{42}=1 \mathrm{y}$ todos los restantes elementos de la cuarta fila lo hacemos cero, usando las propiedades. $|A|=\left|\begin{array}{cccc} -2 & 5 & -5 & 3 \\ 5 & -2 & 7 & 3 \\ 4 & -3 & 6 & 5 \\ -3 & 2 & 2 & 4 \end{array}\right|$

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Problem 270

Calculate $34 \cdot \begin{pmatrix} 2 & 5 \\ 4 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 7 \\ 3 & 0 \end{pmatrix}$ .

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Problem 271

Hitung nilai $P_{31}$ dari $P=3\begin{bmatrix}-1 & -2 \\ 2 & 4 \\ 1 & 5\end{bmatrix}$ .

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Problem 272

Solve using augmented matrix methods. $\begin{array}{rr} 2 x_{1}-x_{2}= & -5 \\ 3 x_{1}+2 x_{2}= & 3 \end{array}$

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Problem 273

Consider the following matrix. $A=\left[\begin{array}{ccc} 2 & 1 & 0 \\ 3 & -1 & 2 \\ 0 & 1 & -1 \end{array}\right]$
Choose the correct description of $A$ . Find $A^{-1}$ if it exists.

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Problem 274

23. Let $A=\left[\begin{array}{rr}-6 & 12 \\ -3 & 6\end{array}\right]$ and $w=\left[\begin{array}{l}2 \\ 1\end{array}\right]$ . Determine if $w$ is in Col Al Is in in Nol A?
24. Let $A=\left[\begin{array}{rrr}-8 & -2 & -9 \\ 6 & 4 & 8 \\ 4 & 0 & 4\end{array}\right]$ and $w=\left[\begin{array}{r}2 \\ 1 \\ -2\end{array}\right]$ . Determine if w is in $\mathrm{Col} A$ . Is w in Nul A?
In Exercises 15 and 16 , find $A$ ouch that the given set is $\operatorname{Col} A$
15. $\left\{\left[\begin{array}{c}2 s+3 t \\ r+s-2 t \\ 4 r+s \\ 3 r-s-t\end{array}\right]: r, s, t\right.$ real $\}$
16. $\left\{\left[\begin{array}{c}b-c \\ 2 b+c+d \\ 5 c-4 d \\ d\end{array}\right]: b, c, d\right.$ real $\}$

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Problem 275

Determine the order of the matrix. $\left[\begin{array}{rr} 5 & 3 \\ 4 & 4 \\ 2 & 3 \\ -5 & 3 \\ 3 & 7 \end{array}\right]$ $\square$ $\times$

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Problem 276

Subtract 3 times the first row from the matrix. Find the new first row: $\left[\begin{array}{cc}-2 & 1\end{array}\right]-3\left[\begin{array}{cc}1 & 0\end{array}\right]$

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Problem 277

Calculate the determinant of the matrix: $\Delta=\left|\begin{array}{llll}1 & 2 & 3 & 4 \\ 3 & 5 & 7 & 9 \\ 4 & 7 & 11 & 15 \\ 8 & 4 & 21 & 30\end{array}\right|$

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Problem 278

\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline & & & & & & & & & & & & \\ \hline \end{tabular}
Circle the correct answer 1) If $A=\left[\begin{array}{cc}2 & 1 \\ 0 & -3\end{array}\right]$ and $p(x)=x^{2}-3 x+1$ , then $p(A)=$ a) $\left[\begin{array}{cc}-1 & 0 \\ 0 & 14\end{array}\right]$ b) $\left[\begin{array}{cc}-1 & -4 \\ 0 & 19\end{array}\right]$ c) $\left[\begin{array}{cc}2 & 1 \\ 0 & -3\end{array}\right]$ d) $\left|\begin{array}{cc}2 & 1 \\ 0 & -3\end{array}\right|$ e) $\left[\begin{array}{cc}1 & -3 \\ 0 & 16\end{array}\right]$ 2) Let $A=\left[\begin{array}{cc}3 & 0 \\ -1 & 2 \\ 1 & 1\end{array}\right], B=\left[\begin{array}{cc}1 & -1 \\ 0 & 2\end{array}\right], C=\left[\begin{array}{ccc}1 & -1 & 2 \\ 3 & 1 & 5\end{array}\right]$ . Then $\left(A+C^{7}\right) B=$ a) $\left[\begin{array}{cc}4 & 2 \\ -2 & 8 \\ 3 & 9\end{array}\right]$ b) $\left[\begin{array}{cc}3 & 3 \\ -2 & 8 \\ 3 & 9\end{array}\right]$ c) $\left[\begin{array}{cc}2 & 4 \\ -2 & 8 \\ 3 & 9\end{array}\right]$ d) $\left[\begin{array}{cc}5 & 1 \\ -2 & 8 \\ 3 & 9\end{array}\right]$ e) $\left[\begin{array}{cc}4 & 2 \\ -2 & 8 \\ 3 & 9\end{array}\right]$ 3) The values of $a$ such that the matrix $A=\left[\begin{array}{ccc}2 & b+c & a-2 b+2 c \\ 0 & -5 & -2 \\ 3 & c-1 & 8\end{array}\right]$ is symmetric are a) $a=-9$ b) $a=11$ c) $a=-5$ d) $a=-1$ e) $a=7$ 4) If a matrix $A$ satisfies $A^{2}+A-I=0$ , then $A^{-1}=$ a) $A-I$ b) $A+I$ c) $I-A$ d) $-I-A$ e) $A$ 5) If $A=\left[\begin{array}{cc}1 & -2 \\ 0 & 1\end{array}\right]$ and $B=\operatorname{tr}(A) \cdot A A^{T}$ , then $\operatorname{det}(B)=$ a) 400 b) 36 c) 4 d) 1 e) 0 6) If $\left|\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right|=-6$ , then $\left|\begin{array}{ccc}a & b & c \\ d & e & f \\ g-4 d & h-4 e & i-4 f\end{array}\right|=$ $\begin{array}{ll}\text { a) } 24 & \text { b) }-24\end{array}$ b) -24 c) 30 d) -30 e) 4 7) The value of $a$ such that the system $-6 x-\left(a^{2}+15\right) y=0$ $x+(a+1) y=0$ has only the trivial solution is a) $a \neq 2$ b) $a=3$ c) $a \neq 3$ d) $\alpha=2$ e) $a=1$ Page 1 of 2

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Problem 279

Find matrix $P$ and $D$ for the following Group 1: $A=\left[\begin{array}{lll}2 & 0 & 4 \\ 0 & 3 & 0 \\ 0 & 0 & 6\end{array}\right]$ Group 2: $B=\left[\begin{array}{ccc}-1 & 2 & 0 \\ 0 & 4 & 1 \\ 0 & 0 & 7\end{array}\right]$

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Problem 280

Find the following matrices where $A=\left[\begin{array}{rr}8 & 6 \\ 3 & 0 \\ -8 & 3\end{array}\right]$ and $B=\left[\begin{array}{rr}-8 & 3 \\ 0 & 6 \\ -5 & -3\end{array}\right]$ . a. $A+B$ b. -4 A c. $-8 A+3 B$ a. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $A+B=$ $\square$ (Simplify your answers.) B. This matrix operation is not possible. b. Select the correct choice below and, if necessary, fill in the answer box to complete your choice $\alpha_{0}$ A. $-4 \mathrm{~A}=$ $\square$ (Simplify your answers.) B. This matrix operation is not possible. c. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
$\square$ A. $-8 A+3 B=\square$ (Simplify your answers.) B. This matrix operation is not possible.

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Problem 281

Suppose that $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{4}$ is such that its action on a vector $\left[\begin{array}{c}x \\ y \\ z\end{array}\right]$ is given below: $T\left[\begin{array}{c} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} x+3 y+3 z \\ -x-2 y-z \\ 2 x+8 y+9 z \\ 3 x+8 y+6 z \end{array}\right]$
Find the matrix $M_{D B}(T)$ that represents $T$ relative to the bases $B$ and $D$ shown below: $\begin{array}{l} B=\left\{\left[\begin{array}{c} 1 \\ -2 \\ 2 \end{array}\right],\left[\begin{array}{c} -2 \\ 2 \\ -2 \end{array}\right],\left[\begin{array}{c} 0 \\ -4 \\ 3 \end{array}\right], D=\left\{\begin{array}{c} 1 \\ -3 \\ 4 \\ 1 \end{array}\right],\left[\begin{array}{c} -1 \\ 4 \\ -5 \\ 0 \end{array}\right],\left[\begin{array}{c} -3 \\ 9 \\ -11 \\ -5 \end{array}\right],\left[\begin{array}{c} 1 \\ -4 \\ 7 \\ -3 \end{array}\right]\right\} \\ M_{D B}(T)=\left[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \end{array}$

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Problem 282

Find a basis for the span of the set of columns of the matrix $A$ . $A=\left[\begin{array}{cccc} 1 & 2 & 2 & 1 \\ 1 & 7 & 7 & 1 \\ 2 & 5 & 6 & -3 \end{array}\right]$
Number of Vectors: 1

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Problem 283

Suppose that $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ is such that its action on a vector $\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$ is given below: $T\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} x-y \\ 2 x-3 y-2 z \\ 2 x-y+z \end{array}\right]$
Find the inverse transformation $T^{-1}$ and give its action on a general vector $\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$ . $T^{-1}\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right]$

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Problem 284

Find the product of matrices $Y=M X$ where $M=\begin{pmatrix}1 & -1 & 0 \\ -1 & 0 & 1 \\ 0 & 1 & -1\end{pmatrix}$ and $X=\begin{pmatrix}5 \\ -3 \\ 10\end{pmatrix}$ .

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Problem 285

Find the matrix $C$ given $A=\begin{bmatrix}7 & -5 \\ 10 & 6\end{bmatrix}$ and $B=\begin{bmatrix}6 & 2 \\ -18 & 14\end{bmatrix}$ , where $C=-2A-B$ .

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Problem 286

Find the product, if possible. $\left[\begin{array}{r} 1 \\ 5 \\ -5 \end{array}\right]\left[\begin{array}{rrr} 1 & 5 & -8 \\ -4 & -1 & 5 \end{array}\right]$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The product is $\square$ B. The product is not defined.

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Problem 287

Suppose that $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{4}$ is such that its action on a vector $\left[\begin{array}{c}x \\ y \\ z\end{array}\right]$ is given below: $T\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} x+3 y+3 z \\ -x-2 y-4 z \\ 2 x+5 y+6 z \\ 3 x+8 y+9 z \end{array}\right]$
Find the matrix $M_{D B}(T)$ that represents $T$ relative to the bases $B$ and $D$ shown below: $\begin{array}{l} \left.B=\left\{\begin{array}{l} 1 \\ 2 \\ 2 \end{array}\right],\left[\begin{array}{c} 1 \\ 3 \\ 1 \end{array}\right],\left[\begin{array}{c} 0 \\ 2 \\ -3 \end{array}\right]\right\} D=\left\{\left[\begin{array}{c} 1 \\ 1 \\ -2 \\ 0 \end{array}\right],\left[\begin{array}{c} -1 \\ 0 \\ 1 \\ -1 \end{array}\right],\left[\begin{array}{c} 1 \\ 1 \\ -3 \\ 1 \end{array}\right],\left[\begin{array}{c} 0 \\ 1 \\ 0 \\ -1 \end{array}\right]\right\} \\ M_{D B}(T)=\left[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \end{array}$

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Problem 288

Dr. Abdullah Shukri
Let the orthogonal matrix $Q$ be: $Q=\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right)$ p 1: Verify Orthogonality 2: Find the eigenvalues p 3: Find the eigenvectors

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Problem 289

You must clearly show your steps for every problem below.
1. Find all distinct (real or complex) eigenvalues of $A$ . Then find the basic eigenvectors of $A$ corresponding to each eigenvalue. $A=\left[\begin{array}{ccc} 4 & 20 & -20 \\ -6 & -24 & 24 \\ -4 & -16 & 16 \end{array}\right]$

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Problem 290

A square matrix $A$ is idempotent if $A^{2}=A$ . Let $V$ be the vector space of all $2 \times 2$ matrices with real entries. Let $H$ be the set of all $2 \times 2$ idempotent matrices with real entries. Is $H$ a subspace of the vector space $V$ ?
1. Does $H$ contain the zero vector of $V$ ? choose
2. Is $H$ closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in $H$ whose sum is not in $H$ , using a comma separated list and syntax such as $[[1,2],[3,4]],[[5,6],[7,8]]$ for the answer $\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right],\left[\begin{array}{ll}5 & 6 \\ 7 & 8\end{array}\right]$ . (Hint: to show that $H$ is not closed under addition, it is sufficient to find two idempotent matrices $A$ and $B$ such that $(A+B)^{2} \neq(A+B)$ .) $\square$
3. Is $H$ closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in $\mathbb{R}$ and a matrix in $H$ whose product is not in $H$ , using a comma separated list and syntax such as $2,[[3,4],[5,6]]$ for the answer $2,\left[\begin{array}{ll}3 & 4 \\ 5 & 6\end{array}\right]$ . (Hint: to show that $H$ is not closed under scalar multiplication, it is sufficient to find a real number $r$ and an idempotent matrix $A$ such that $(r A)^{2} \neq(r A)$ .) $\square$
4. Is $H$ a subspace of the vector space $V$ ? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3. choose

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Problem 291

Find a basis for the column space of $A=\left[\begin{array}{cccc} 4 & 2 & 0 & -1 \\ -4 & -4 & -3 & 1 \\ 4 & 2 & 0 & -1 \end{array}\right]$

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Problem 292

To find the null space of the matrix
$\begin{bmatrix} 1 & 7 & -2 & 14 & 0 \\ 3 & 0 & 1 & -2 & 3 \\ 6 & 1 & -1 & 0 & 4 \end{bmatrix}$
and express it as $\operatorname{span}\{A, B\}$ , where $A$ and $B$ are vectors that form a basis for the null space.

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Problem 293

|EXERCISE SET 1.2
Reduced Echelon Form of a Matrix
1. Determine whether the following matrices are in reduced echelon form. If a matrix is not in reduced echelon form give a reason.
(a) $\left[\begin{array}{rrrr}1 & 0 & 3 & -2 \\ 0 & 0 & 1 & 8 \\ 0 & 1 & 4 & 9\end{array}\right]$ (b) $\left[\begin{array}{lllll}1 & 2 & 0 & 0 & 4 \\ 0 & 0 & 1 & 0 & 6 \\ 0 & 0 & 0 & 1 & 5\end{array}\right]$ $\left[\begin{array}{lllll}1 & 0 & 4 & 2 & 6 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 1\end{array}\right]$ (e) $\left[\begin{array}{lllll}1 & 0 & 2 & 0 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 2 & 0 & 7 \\ 0 & 0 & 0 & 1 & 3\end{array}\right]$ (f) $\left[\begin{array}{lllll}1 & 0 & 4 & 0 & 0 \\ 0 & 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1\end{array}\right]$ (g) $\left[\begin{array}{lllll}1 & 0 & 0 & 5 & 3 \\ 0 & 0 & 1 & 0 & 3 \\ 0 & 1 & 2 & 3 & 7\end{array}\right]$ (h) $\left[\begin{array}{lllll}0 & 0 & 1 & 0 & 4 \\ 0 & 0 & 0 & 1 & 5 \\ 0 & 1 & 0 & 0 & 3\end{array}\right]$ (i) $\left[\begin{array}{rrrrr}1 & 5 & -3 & 0 & 7 \\ 0 & 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]$

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Problem 294

Question Computer $A^{T}$ for the matrix below. $A=\left[\begin{array}{lll} 1 & 4 & 2 \\ 3 & 5 & 0 \\ 1 & 2 & 3 \end{array}\right]$
Provide your answer below:

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Problem 295

SORULARI QUESTIONS
1. For given vectors $u=(-2,4,0) v=(1,-3,6)$ a) Find $\cos \theta$ between two vectors. (10p) b) When does the dot product give zero? (Sp) c) Find the distance two vectors. ? (Sp)
2. Find the solution of the linear equation system by Gauss Jordan Elimination Method. (400) $\begin{aligned} 3 x_{1}+x_{2}+x_{3} & =12 \\ 2 x_{1}-x_{2}+2 x_{3} & =9 \\ x_{1}+x_{2}+x_{3} & =6 \end{aligned}$
3. $\boldsymbol{A}=\left[\begin{array}{rrrr}1 & -2 & 0 & 3 \\ 0 & 1 & 5 & 4 \\ 2 & -1 & -2 & 2 \\ 3 & 4 & 1 & -3\end{array}\right]$
Find the $\mathrm{A}=$ L.D.U Factorization ( 40 p )

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Problem 296

A, B and C are three different matrices. The order $m \times n$ of the $A$ matrix is $2 \times 3$ . In order for the operation $A \times(B-C)$ to be possible, matrices $B$ and $C$ cannot be the order of a. $3 \times 2$ b. $3 \times 3$ c. $3 \times 1$ d. $2 \times 2$

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Problem 297

uizzes ng an external tool Quiz \#12
Given $\bar{p}=1 x+2 x^{2}$ , find the norm of $p$ .
2
This question accepts numbers or formulas. Help I Switch to Equation Editor | Preview
Given the math $U=\left[\begin{array}{cc}4 & 3 \\ -4 & -1\end{array}\right]$ find the norm of Stant

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Problem 298

Consider the inner product on $M_{22}$ defined by $<U, V>=u_{1} v_{1}+u_{2} v_{2}+u_{3} v_{3}+u_{4} v_{4}$ where $U=\left[\begin{array}{ll}u_{1} & u_{2} \\ u_{3} & u_{4}\end{array}\right]$ and $v=\left[\begin{array}{ll}v_{1} & v_{2} \\ v_{3} & v_{4}\end{array}\right]$ Using this inner product, the matrices $\left[\begin{array}{cc}-2 & 2 \\ -4 & -4\end{array}\right]$ and $\left[\begin{array}{cc}1 & 1 \\ -3 & 3\end{array}\right]$ are orthogonal True False

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Problem 299

QUESTION 9 Write a system of linear equations in three variables, and then use matrices to solve the system. There were approximately 100,000 vehicles sold at a particular dealership last year. The dealer tracks sales by age group for marketing purposes. The percentage of 36 - to 59 -year-old buyers and the percentage of buyers 60 and older combined exceeds the percentage of buyers 35 and younger by $38 \%$ . If the percentage of buyers in the oldest group is doubled, it is $24 \%$ less than the percentage of users in the middle group. Find the percentage of buyers in each of the three age groups. 15\% 35 and younger; 54\% 36-59 year olds; 31\% 60 and older 25\% 35 and younger, 56\% 36-59 year olds; 19\% 60 and older $33 \% 35$ and younger; 51\% 36-59 year olds; 16\% 60 and older 31\% 35 and younger; 54\% 36-59 year olds; 15\% 60 and older

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Problem 300

$\text { Let } A=\left[\begin{array}{ccc} 0 & 0 & 4 \\ 0 & -4 & 0 \\ -2 & 0 & -6 \end{array}\right]$ (a) Determine the eigenvalues $\lambda_{1}$ and $\lambda_{2}$ of $A$ where $\lambda_{1}<\lambda_{2}$ . $\lambda_{1}=$ $\square$ $\lambda_{2}=$ $\square$ (b) Determine the algebraic multiplicity $m$ of each of the eigenvalues in part (a). $m\left(\lambda_{1}\right)=$ $\square$ $m\left(\lambda_{2}\right)=$ $\square$

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Matrices

Problem 201

Problem 202

Problem 203

Problem 204

Problem 205

Problem 206

11. Galoulate the determinant. [5231]\left[\begin{array}{ll} 5 & 2 \\ 3 & 1 \end{array}\right][53​21​] 1 11 0 −11-11−11 −1-1−1

Problem 207

Problem 208

Calculate the products of matrices SSS and TTT: STS TST and TST STS to show multiplication is not commutative. Fill in the boxes.

Problem 209

Find the product matrix PQP QPQ for the matrices P=[271704−1−3]P=\begin{bmatrix} 2 & 7 \\ 1 & 7 \\ 0 & 4 \\ -1 & -3 \end{bmatrix}P=⎣⎡​210−1​774−3​⎦⎤​ and Q=[−2−3]Q=\begin{bmatrix} -2 \\ -3 \end{bmatrix}Q=[−2−3​].

Problem 210

Find the matrix result of m×Hm \times Hm×H given the equations involving matrices and constants.

Problem 211

Find the product matrix BCB CBC for matrices B=(−2305)B=\begin{pmatrix}-2 & 3 \\ 0 & 5\end{pmatrix}B=(−20​35​) and C=(04−1301)C=\begin{pmatrix}0 & 4 & -1 \\ 3 & 0 & 1\end{pmatrix}C=(03​40​−11​).

Problem 212

Show that matrix multiplication is not commutative by calculating STS TST and TST STS for the matrices S=[41−3−2]S=\begin{bmatrix} 4 & 1 \\ -3 & -2 \end{bmatrix}S=[4−3​1−2​] and T=[04−43]T=\begin{bmatrix} 0 & 4 \\ -4 & 3 \end{bmatrix}T=[0−4​43​].

Problem 213

Find the dimensions of the product matrix formed by multiplying any two of the following matrices: AAA, BBB, CCC, DDD.

Problem 214

Find the matrix resulting from m×Hm \times Hm×H given the equations involving matrices.

Problem 215

Find the general solution for the system with the matrix: [1412416−1−17] \left[\begin{array}{rrrr} 1 & 4 & 1 & 2 \\ 4 & 16 & -1 & -17 \end{array}\right] [14​416​1−1​2−17​] Choose A, B, C, or D for the solution type.

Problem 216

Find the values of hhh for which the matrix is consistent: [1h4−36−16] \begin{bmatrix} 1 & h & 4 \\ -3 & 6 & -16 \end{bmatrix} [1−3​h6​4−16​] Choose A, B, C, or D.

Problem 217

Row reduce the matrix and find pivot positions. Given matrix: [123456786789] \left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \end{array}\right] ⎣⎡​156​267​378​489​⎦⎤​

Problem 218

Row reduce the matrix below to reduced echelon form and identify pivot positions.[123456786789] \left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \end{array}\right] ⎣⎡​156​267​378​489​⎦⎤​

Problem 219

Problem 220

Find values of h for which the matrix is consistent: [12−45h−20] \left[\begin{array}{rrr} 1 & 2 & -4 \\ 5 & h & -20 \end{array}\right] [15​2h​−4−20​] Choose A, B, C, or D.

Problem 221

Find the general solution for the system with the augmented matrix: [12033505] \left[\begin{array}{llll} 1 & 2 & 0 & 3 \\ 3 & 5 & 0 & 5 \end{array}\right] [13​25​00​35​] Choose the correct option for x1x_1x1​, x2x_2x2​, and x3x_3x3​.

Problem 222

Problem 223

Solve the system from the matrix: [01−361−37−14] \begin{bmatrix} 0 & 1 & -3 & 6 \\ 1 & -3 & 7 & -14 \end{bmatrix} [01​1−3​−37​6−14​] Choose the correct solution set or state if there's no solution.

Problem 224

Find values of hhh for which the matrix is consistent: [14−32h−6] \left[\begin{array}{rrr} 1 & 4 & -3 \\ 2 & h & -6 \end{array}\right] [12​4h​−3−6​] Choices: A. h≠h \neqh= B. h=h=h= C. for all hhh D. for no hhh.

Problem 225

Problem 226

Find the general solution for the system with the matrix: [01−221−46−5] \left[\begin{array}{rrrr} 0 & 1 & -2 & 2 \\ 1 & -4 & 6 & -5 \end{array}\right] [01​1−4​−26​2−5​] Choose A, B, C, or D for your answer.

Problem 227

Find the general solution for the system with the augmented matrix: [110−3210−9] \left[\begin{array}{llll} 1 & 1 & 0 & -3 \\ 2 & 1 & 0 & -9 \end{array}\right] [12​11​00​−3−9​] Choose A, B, C, or D based on the solution.

Problem 228

Solve the system by using the inverse of the coefficient matrix. −5x−y+2z=44x+9y−5z=−122x−5y+z=2\begin{array}{l} -5 x-y+2 z=4 \\ 4 x+9 y-5 z=-12 \\ 2 x-5 y+z=2 \end{array}−5x−y+2z=44x+9y−5z=−122x−5y+z=2​

Problem 229

Problem 230

Problem 231

Problem 232

Problem 233

Find the general solution of the system x′=(−211−2)x+(2e−t3t)=Ax+g(t)\mathbf{x}^{\prime}=\left(\begin{array}{rr} -2 & 1 \\ 1 & -2 \end{array}\right) \mathbf{x}+\binom{2 e^{-t}}{3 t}=\mathbf{A} \mathbf{x}+\mathbf{g}(t)x′=(−21​1−2​)x+(3t2e−t​)=Ax+g(t)

Problem 234

Problem 235

Problem 236

Problem 237

Determine the rank of the matrix [1−35−39−152−610]\left[\begin{array}{ccc}1 & -3 & 5 \\ -3 & 9 & -15 \\ 2 & -6 & 10\end{array}\right]⎣⎡​1−32​−39−6​5−1510​⎦⎤​ 304 2

Problem 238

Problem 239

If AAA is a 3×33 \times 33×3 matrix such that det⁡(A)=7\operatorname{det}(A)=7det(A)=7, then: det⁡(2ATA−1)=\operatorname{det}\left(2 A^{T} A^{-1}\right)=det(2ATA−1)=Select one: a. 56 b. 8 c. 98 d. 14

Problem 240

Evaluate the determinant of the matrix. E=[4070]E=\left[\begin{array}{ll} 4 & 0 \\ 7 & 0 \end{array}\right]E=[47​00​]The determinant is □\square□ .

Problem 241

If AAA is a 3×33 \times 33×3 matrix such that det⁡(A)=3\operatorname{det}(A)=3det(A)=3 then det⁡(A2)=6\operatorname{det}\left(A^{2}\right)=6det(A2)=6.Select one: True False

Problem 242

If the adjoint of amatrix is (2−345)\left(\begin{array}{cc}2 & -3 \\ 4 & 5\end{array}\right)(24​−35​) find the matrix.

Problem 243

Problem 244

Problem 245

Problem 246

Problem 247

Suppose that AAA is an n×mn \times mn×m matrix of rank 4 , the nullity of AAA is 2 , and the column space of AAA is a subspace of R8\mathbf{R}^{8}R8. Find the dimensions of AAA. AAA has □\square□ rows and □\square□ columns.

Problem 248

Problem 249

Simplify. −14[12−2028−12]-\frac{1}{4}\left[\begin{array}{ll} 12 & -20 \\ 28 & -12 \end{array}\right]−41​[1228​−20−12​]

Problem 250

Problem 251

Find the values of hhh for which the matrix is consistent: [12−35h−15] \left[\begin{array}{rrr} 1 & 2 & -3 \\ 5 & h & -15 \end{array}\right] [15​2h​−3−15​] Options: A. h≠h \neqh= B. h=h=h= C. Consistent for all hhh D. Not consistent for any hhh.

11. Galoulate the determinant. $\left[\begin{array}{ll} 5 & 2 \\ 3 & 1 \end{array}\right]$ 1 11 0 $-11$ $-1$

Calculate the products of matrices $S$ and $T$ : $S T$ and $T S$ to show multiplication is not commutative. Fill in the boxes.

Find the product matrix $P Q$ for the matrices $P=\begin{bmatrix} 2 & 7 \\ 1 & 7 \\ 0 & 4 \\ -1 & -3 \end{bmatrix}$ and $Q=\begin{bmatrix} -2 \\ -3 \end{bmatrix}$ .

Find the matrix result of $m \times H$ given the equations involving matrices and constants.

Find the product matrix $B C$ for matrices $B=\begin{pmatrix}-2 & 3 \\ 0 & 5\end{pmatrix}$ and $C=\begin{pmatrix}0 & 4 & -1 \\ 3 & 0 & 1\end{pmatrix}$ .

Show that matrix multiplication is not commutative by calculating $S T$ and $T S$ for the matrices $S=\begin{bmatrix} 4 & 1 \\ -3 & -2 \end{bmatrix}$ and $T=\begin{bmatrix} 0 & 4 \\ -4 & 3 \end{bmatrix}$ .

Find the dimensions of the product matrix formed by multiplying any two of the following matrices: $A$ , $B$ , $C$ , $D$ .

Find the matrix resulting from $m \times H$ given the equations involving matrices.

Find the general solution for the system with the matrix:
$\left[\begin{array}{rrrr} 1 & 4 & 1 & 2 \\ 4 & 16 & -1 & -17 \end{array}\right]$
Choose A, B, C, or D for the solution type.

Find the values of $h$ for which the matrix is consistent: $\begin{bmatrix} 1 & h & 4 \\ -3 & 6 & -16 \end{bmatrix}$ Choose A, B, C, or D.

Row reduce the matrix and find pivot positions. Given matrix: $\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \end{array}\right]$

Row reduce the matrix below to reduced echelon form and identify pivot positions.
$\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \end{array}\right]$

Find values of h for which the matrix is consistent: $\left[\begin{array}{rrr} 1 & 2 & -4 \\ 5 & h & -20 \end{array}\right]$ Choose A, B, C, or D.

Find the general solution for the system with the augmented matrix:
$\left[\begin{array}{llll} 1 & 2 & 0 & 3 \\ 3 & 5 & 0 & 5 \end{array}\right]$
Choose the correct option for $x_1$ , $x_2$ , and $x_3$ .

Solve the system from the matrix: $\begin{bmatrix} 0 & 1 & -3 & 6 \\ 1 & -3 & 7 & -14 \end{bmatrix}$ Choose the correct solution set or state if there's no solution.

Find values of $h$ for which the matrix is consistent: $\left[\begin{array}{rrr} 1 & 4 & -3 \\ 2 & h & -6 \end{array}\right]$ Choices: A. $h \neq$ B. $h=$ C. for all $h$ D. for no $h$ .

Find the general solution for the system with the matrix:
$\left[\begin{array}{rrrr} 0 & 1 & -2 & 2 \\ 1 & -4 & 6 & -5 \end{array}\right]$
Choose A, B, C, or D for your answer.

Find the general solution for the system with the augmented matrix:
$\left[\begin{array}{llll} 1 & 1 & 0 & -3 \\ 2 & 1 & 0 & -9 \end{array}\right]$
Choose A, B, C, or D based on the solution.

Solve the system by using the inverse of the coefficient matrix. $\begin{array}{l} -5 x-y+2 z=4 \\ 4 x+9 y-5 z=-12 \\ 2 x-5 y+z=2 \end{array}$

Find the general solution of the system $\mathbf{x}^{\prime}=\left(\begin{array}{rr} -2 & 1 \\ 1 & -2 \end{array}\right) \mathbf{x}+\binom{2 e^{-t}}{3 t}=\mathbf{A} \mathbf{x}+\mathbf{g}(t)$

Determine the rank of the matrix $\left[\begin{array}{ccc}1 & -3 & 5 \\ -3 & 9 & -15 \\ 2 & -6 & 10\end{array}\right]$ 3
0
4 2

If $A$ is a $3 \times 3$ matrix such that $\operatorname{det}(A)=7$ , then: $\operatorname{det}\left(2 A^{T} A^{-1}\right)=$
Select one: a. 56 b. 8 c. 98 d. 14

Evaluate the determinant of the matrix. $E=\left[\begin{array}{ll} 4 & 0 \\ 7 & 0 \end{array}\right]$
The determinant is $\square$ .

If $A$ is a $3 \times 3$ matrix such that $\operatorname{det}(A)=3$ then $\operatorname{det}\left(A^{2}\right)=6$ .
Select one: True False

If the adjoint of amatrix is $\left(\begin{array}{cc}2 & -3 \\ 4 & 5\end{array}\right)$ find the matrix.

Suppose that $A$ is an $n \times m$ matrix of rank 4 , the nullity of $A$ is 2 , and the column space of $A$ is a subspace of $\mathbf{R}^{8}$ . Find the dimensions of $A$ . $A$ has $\square$ rows and $\square$ columns.

Simplify. $-\frac{1}{4}\left[\begin{array}{ll} 12 & -20 \\ 28 & -12 \end{array}\right]$

Find the values of $h$ for which the matrix is consistent:
$\left[\begin{array}{rrr} 1 & 2 & -3 \\ 5 & h & -15 \end{array}\right]$
Options: A. $h \neq$ B. $h=$ C. Consistent for all $h$ D. Not consistent for any $h$ .

Find the general solution for the system with the matrix:
$\left[\begin{array}{rrrr} 1 & 1 & 0 & -7 \\ 2 & 1 & 0 & -16 \end{array}\right]$
Select A, B, C, or D.

Find the general solution for the system from the augmented matrix: $\begin{bmatrix} 0 & 1 & -2 & 5 \\ 1 & -4 & 4 & -14 \end{bmatrix}$ Choose A, B, C, or D.

Find the inverse of the $3 \times 3$ matrix $A$ $A=\left[\begin{array}{lll} 1 & 2 & 1 \\ 9 & 1 & 0 \\ 3 & 0 & 1 \end{array}\right]$

Use matrices to solve the system of equations (if possible). Use $x$ and $y$ in terms of the real number a.) $\left\{\begin{array}{r} x+2 y=0 \\ x+y=7 \\ 3 x-2 y=16 \end{array}\right.$

the inverse of $A=\left(\begin{array}{cc} 3 & -5 \\ -2 & 3 \end{array}\right)$

What are the dimensions of this matrix? $\left[\begin{array}{lll} 3 & 6 & 9 \end{array}\right]$ $\square$ by

وجد معكوس المصفوفة A $A=\left[\begin{array}{lll} 2 & 0 & 3 \\ 3 & 1 & 4 \\ 1 & 2 & 5 \end{array}\right]$

Question 1 (1 point) (CAS1U2Q1) Find $\operatorname{det}\left[\begin{array}{cccc}1 & 3 & 0 & -1 \\ 2 & -1 & 2 & 1 \\ 3 & 2 & 0 & 1 \\ -1 & 2 & 1 & 1\end{array}\right]$ 0 $-47$ 23 47 $-23$

4) $-5\left[\begin{array}{cc}-3 & 0 \\ 0 & 5\end{array}\right]$

Finding the determinant of a $3 \times 3$ matrix
Evaluate the following determinant. $\left|\begin{array}{ccc} 0 & -4 & 6 \\ 3 & 3 & -3 \\ -3 & 3 & 5 \end{array}\right|$

Calculate $34 \cdot \begin{pmatrix} 2 & 5 \\ 4 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 7 \\ 3 & 0 \end{pmatrix}$ .

Hitung nilai $P_{31}$ dari $P=3\begin{bmatrix}-1 & -2 \\ 2 & 4 \\ 1 & 5\end{bmatrix}$ .

Solve using augmented matrix methods. $\begin{array}{rr} 2 x_{1}-x_{2}= & -5 \\ 3 x_{1}+2 x_{2}= & 3 \end{array}$

Consider the following matrix. $A=\left[\begin{array}{ccc} 2 & 1 & 0 \\ 3 & -1 & 2 \\ 0 & 1 & -1 \end{array}\right]$
Choose the correct description of $A$ . Find $A^{-1}$ if it exists.