Problem 46 - Precalculus with Limits

Question

In Exercises \(41-46,\) use the matrix capabilities of a graphing utility to find \(A B,\) if possible. $$ A=\left[\begin{array}{rr}{16} & {-18} \\ {-4} & {13} \\ {-9} & {21}\end{array}\right], \quad B=\left[\begin{array}{rrr}{-7} & {20} & {-1} \\\ {7} & {15} & {26}\end{array}\right] $$

Step-by-Step Solution

Answer

The product of the matrices A and B, denoted AB, is \[ \left[\begin{array}{r} -418 & 50 & -468 \ 77 & 5 & 326 \ 84 & 165 & 513 \end{array}\right]. \]

Step 1: Verify the Possibility of Matrix Multiplication

Given matrices A and B where A is a 3x2 matrix and B is a 2x3 matrix. Matrix multiplication is possible if the number of columns of the first matrix equals the number of rows of the second matrix. So, here for matrices A and B, 2 columns in A equal 2 rows in B, hence multiplication AB is possible.

Step 2: Perform the Matrix Multiplication

The multiplication of two matrices is done element-wise. Each element of the result matrix is computed by multiplying the elements of the rows of the first matrix with the corresponding elements of the columns of the second matrix and then adding them. The (i, j)th entry of the result matrix AB, will be calculated by taking the dot product of the ith row of A with the jth column of B. Using these rules, obtain the resultant product matrix AB: \(AB=\left[\begin{array}{r} 16 & -18 \ -4 & 13 \ -9 & 21\end{array}\right] \times \left[\begin{array}{r} -7 & 20 & -1 \ 7 & 15 & 26\end{array}\right] = \left[\begin{array}{r} -418 & 50 & -468 \ 77 & 5 & 326 \ 84 & 165 & 513 \end{array}\right]\)