Problem 41 - Precalculus with Limits

Question

In Exercises \(41-46,\) use the matrix capabilities of a graphing utility to find \(A B,\) if possible. $$ A=\left[\begin{array}{rrr}{7} & {5} & {-4} \\ {-2} & {5} & {1} \\ {10} & {-4} & {-7}\end{array}\right], \quad B=\left[\begin{array}{rrr}{2} & {-2} & {3} \\ {8} & {1} & {4} \\ {-4} & {2} & {-8}\end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix after multiplication, \(C\), is \[ \left[ \begin{array}{ccc} 62 & 19 & 22 \ 36 & 0 & 8 \ 0 & -28 & 62 \end{array} \right] \]

Step 1: Verify if multiplication is possible

Check if the number of columns of matrix \(A\) equals the number of rows of matrix \(B\). Both matrices are of size 3x3, so multiplication is possible.

Step 2: Set up Placeholders

Establish a placeholder matrix, which will have identical dimensions to the resulting matrix. Since our matrices are both 3x3 the resulting matrix will also be 3x3. Set up placeholders with indices corresponding to row of first matrix and column of second matrix. Rename it to \(C\) for this calculation's convenience.

Step 3: Process the multiplication row by column

Process the multiplication row by column. To calculate the entry at an index \(i, j\), take the \(i\)th row of the first matrix (\(A\)) and the \(j\)th column of the second matrix (\(B\)) and perform an element-wise multiplication followed by a summation. Do this for every indices pair in matrix \(C\).

Step 4: Calculate all elements of the resulting matrix

Calculate all elements of the resulting matrix, filling in the placeholders of matrix \(C\). For example, the first entry of the resulting matrix would be (7*2 + 5*8 + -4*-4), likewise, calculate the rest of the entries according to the rules specified in Step 3.

Step 5: Finalize the solution

Write down the results obtained in Step 4 into the placeholders of matrix \(C\), which gives the final multiplication result.