Problem 42 - Precalculus with Limits

Question

In Exercises \(41-46,\) use the matrix capabilities of a graphing utility to find \(A B,\) if possible. $$ A=\left[\begin{array}{rrr}{11} & {-12} & {4} \\ {14} & {10} & {12} \\ {6} & {-2} & {9}\end{array}\right], \quad B=\left[\begin{array}{rr}{12} & {10} \\\ {-5} & {12} \\ {15} & {16}\end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix AB after matrix multiplication will be a 3x2 matrix with individual values calculated by dot product method (multiplying and adding corresponding entries in row of A and column of B).

Step 1: Identify the matrices and check conditions for multiplication

Matrix A is a 3x3 matrix and matrix B is a 3x2 matrix. For matrix multiplication, the number of columns in the first matrix (A) should be equal to the number of rows in the second matrix (B). Here, A has 3 columns and B has 3 rows, so multiplication is possible.

Step 2: Start multiplying row by column

For this, the first row of A should be multiplied with the first column of B. This will be the (1,1) entry of the result matrix AB. Then, multiply the first row of A with the second column of B (which will be the (1,2) entry), and so on.

Step 3: Continue multiplying row by column

Repeat Step 2 for the remaining rows of matrix A. This means that you will then multiply the second row of A with every column of B (resulting in the second row of the result matrix), and the third row of A with every column of B (resulting in the third row of the result matrix).

Step 4: Final calculation of matrix AB

After the multiplication is finished, add the multiplication results to get the final matrix AB, which should be a 3x2 matrix.