Find (if possible) a. AB and b. BA \[ A=\left[\begin{array}{rr} -8 & 6 \\ -5 & 1 \end{array}\right], B=\left[\begin{array}{rr} 5 & -8 \\ -1 & 7 \end{array}\right] \]
Solution
Step 1 ：We are given two matrices A and B, where A = \(\begin{bmatrix} -8 & 6 \\ -5 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 5 & -8 \\ -1 & 7 \end{bmatrix}\).
Step 2 ：We are asked to find the products of these matrices in two orders: AB and BA. Note that matrix multiplication is not commutative, so the order in which the matrices are multiplied can affect the result.
Step 3 ：To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In this case, both matrices are 2x2, so they can be multiplied in either order.
Step 4 ：The product of two matrices is calculated by taking the dot product of the rows of the first matrix with the columns of the second matrix. For example, the element in the first row and first column of the product matrix is calculated by multiplying the elements of the first row of the first matrix by the corresponding elements of the first column of the second matrix and adding the results.
Step 5 ：Using this method, we find that the product of the matrices A and B in the order AB is \(\begin{bmatrix} -46 & 106 \\ -26 & 47 \end{bmatrix}\).
Step 6 ：Similarly, the product of the matrices A and B in the order BA is \(\begin{bmatrix} 0 & 22 \\ -27 & 1 \end{bmatrix}\).
Step 7 ：Thus, the final answer is \(\boxed{AB = \begin{bmatrix} -46 & 106 \\ -26 & 47 \end{bmatrix}}\) and \(\boxed{BA = \begin{bmatrix} 0 & 22 \\ -27 & 1 \end{bmatrix}}\).
