Problem 43 - Precalculus with Limits

Question

In Exercises \(41-46,\) use the matrix capabilities of a graphing utility to find \(A B,\) if possible. $$ A=\left[\begin{array}{rrrr}{-3} & {8} & {-6} & {8} \\ {-12} & {15} & {9} & {6} \\ {5} & {-1} & {1} & {5}\end{array}\right], \quad B=\left[\begin{array}{rrr}{3} & {1} & {6} \\ {24} & {15} & {14} \\ {16} & {10} & {21} \\ {8} & {-4} & {10}\end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix after multiplying A and B is a 3x3 matrix, with the entries calculated using the dot product of the corresponding rows from A and columns from B.

Step 1: Set Up Matrix Multiplication

Set up the matrix multiplication operation, keeping in mind that matrix multiplication is not commutative, so the order of the matrices (A times B) matters.

Step 2: Calculate Entries

Calculate each entry in the resulting matrix by taking the dot product of the corresponding row from the first matrix (A) and the corresponding column from the second matrix (B). The dot product is calculated by multiplying corresponding entries and then adding those products together. For example, the entry in the first row and first column of the result will be (-3*3) + (8*24) + (-6*16) + (8*8). Repeat this process for each entry in the result matrix.

Step 3: Finalize Matrix

After calculating all the entries, finalize the resulting matrix. This will be a 3x3 matrix, as determined by the number of rows in the first matrix (A) and the number of columns in the second matrix (B).