QuestionIf possible, find $A B$ . (If not possible, enter IMPOSSIBLE in any cell of the matrix.) $\begin{array}{c} A=\left[\begin{array}{rr} -1 & 9 \\ -4 & 5 \\ 0 & 4 \end{array}\right], \quad B=\left[\begin{array}{ll} 3 & 4 \\ 0 & 9 \end{array}\right] \\ A B=-7 c \\ -12 \end{array} \begin{array}{cc} 77 & \Rightarrow \\ -39 \end{array}$ -12 $\square$ 77 $\square$ 29

Studdy Solution

STEP 1

1. Matrix multiplication is defined when the number of columns in the first matrix equals the number of rows in the second matrix.
2. We will check the dimensions of matrices $A$ and $B$ to determine if multiplication is possible.
3. If multiplication is possible, we will compute the product $AB$ .

STEP 2

1. Check the dimensions of matrices $A$ and $B$ .
2. Determine if matrix multiplication is possible.
3. Compute the product $AB$ if possible.

STEP 3

First, determine the dimensions of matrices $A$ and $B$ .
Matrix $A$ is a $3 \times 2$ matrix: $A = \begin{bmatrix} -1 & 9 \\ -4 & 5 \\ 0 & 4 \end{bmatrix}$
Matrix $B$ is a $2 \times 2$ matrix: $B = \begin{bmatrix} 3 & 4 \\ 0 & 9 \end{bmatrix}$

STEP 4

Check if the number of columns in $A$ (which is 2) is equal to the number of rows in $B$ (which is 2). Since they are equal, matrix multiplication is possible.

STEP 5

Compute the product $AB$ . The resulting matrix will have dimensions $3 \times 2$ .
To find each element of the resulting matrix $AB$ , use the formula for matrix multiplication:
$(AB)_{ij} = \sum_{k=1}^{n} A_{ik} \cdot B_{kj}$
Where $n$ is the number of columns in $A$ (or rows in $B$ ).
Compute each element:
1. First row, first column: $ (AB)_{11} = (-1)(3) + (9)(0) = -3 + 0 = -3 \]
2. First row, second column: $ (AB)_{12} = (-1)(4) + (9)(9) = -4 + 81 = 77 \]
3. Second row, first column: $ (AB)_{21} = (-4)(3) + (5)(0) = -12 + 0 = -12 \]
4. Second row, second column: $ (AB)_{22} = (-4)(4) + (5)(9) = -16 + 45 = 29 \]
5. Third row, first column: $ (AB)_{31} = (0)(3) + (4)(0) = 0 + 0 = 0 \]
6. Third row, second column: $ (AB)_{32} = (0)(4) + (4)(9) = 0 + 36 = 36 \]
Thus, the product $AB$ is:
$AB = \begin{bmatrix} -3 & 77 \\ -12 & 29 \\ 0 & 36 \end{bmatrix}$
The resulting matrix $AB$ is:
$\begin{bmatrix} -3 & 77 \\ -12 & 29 \\ 0 & 36 \end{bmatrix}$

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