QuestionIf possible, find $A B, B A$ , and $A^{2}$ . (If not possible, enter IMPOSSIBLE in any cell of the matrix.) $A=\left[\begin{array}{ll} 1 & 4 \\ 3 & 4 \end{array}\right], B=\left[\begin{array}{rr} 2 & -1 \\ -1 & 6 \end{array}\right]$ (a) $A B$ $A B=\left[\begin{array}{cc} \square & \square \\ \square & \square \end{array}\right] \Rightarrow$ (b) $B A$ (c) $A^{2}$

Studdy Solution

STEP 1

1. Matrix multiplication is defined only when the number of columns in the first matrix equals the number of rows in the second matrix.
2. The matrices $A$ and $B$ are both $2 \times 2$ matrices.
3. The product $A B$ is possible if the number of columns in $A$ equals the number of rows in $B$ .
4. The product $B A$ is possible if the number of columns in $B$ equals the number of rows in $A$ .
5. The square of a matrix $A^2$ is possible if $A$ is a square matrix.

STEP 2

1. Calculate $A B$ .
2. Calculate $B A$ .
3. Calculate $A^2$ .

STEP 3

To find $A B$ , multiply matrix $A$ by matrix $B$ :
$A = \begin{bmatrix} 1 & 4 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & -1 \\ -1 & 6 \end{bmatrix}$
The product $A B$ is given by:
$A B = \begin{bmatrix} (1)(2) + (4)(-1) & (1)(-1) + (4)(6) \\ (3)(2) + (4)(-1) & (3)(-1) + (4)(6) \end{bmatrix}$
Calculating each element:
$A B = \begin{bmatrix} 2 - 4 & -1 + 24 \\ 6 - 4 & -3 + 24 \end{bmatrix}$ $A B = \begin{bmatrix} -2 & 23 \\ 2 & 21 \end{bmatrix}$

STEP 4

To find $B A$ , multiply matrix $B$ by matrix $A$ :
$B = \begin{bmatrix} 2 & -1 \\ -1 & 6 \end{bmatrix}, \quad A = \begin{bmatrix} 1 & 4 \\ 3 & 4 \end{bmatrix}$
The product $B A$ is given by:
$B A = \begin{bmatrix} (2)(1) + (-1)(3) & (2)(4) + (-1)(4) \\ (-1)(1) + (6)(3) & (-1)(4) + (6)(4) \end{bmatrix}$
Calculating each element:
$B A = \begin{bmatrix} 2 - 3 & 8 - 4 \\ -1 + 18 & -4 + 24 \end{bmatrix}$ $B A = \begin{bmatrix} -1 & 4 \\ 17 & 20 \end{bmatrix}$

STEP 5

To find $A^2$ , multiply matrix $A$ by itself:
$A = \begin{bmatrix} 1 & 4 \\ 3 & 4 \end{bmatrix}$
The product $A^2$ is given by:
$A^2 = \begin{bmatrix} (1)(1) + (4)(3) & (1)(4) + (4)(4) \\ (3)(1) + (4)(3) & (3)(4) + (4)(4) \end{bmatrix}$
Calculating each element:
$A^2 = \begin{bmatrix} 1 + 12 & 4 + 16 \\ 3 + 12 & 12 + 16 \end{bmatrix}$ $A^2 = \begin{bmatrix} 13 & 20 \\ 15 & 28 \end{bmatrix}$
The products are: (a) $A B = \begin{bmatrix} -2 & 23 \\ 2 & 21 \end{bmatrix}$ (b) $B A = \begin{bmatrix} -1 & 4 \\ 17 & 20 \end{bmatrix}$ (c) $A^2 = \begin{bmatrix} 13 & 20 \\ 15 & 28 \end{bmatrix}$

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