Question

Find AB. Do not use the matrix function in a calculator - do this by hand and show all work.

Answer 1

Given that matrices A, B, C as follows:

`\begin{gathered} A=\begin{bmatrix}{3} & {4} & {} \\ {-5} & {2} & {} \\ {1} & {0} & {}\end{bmatrix} \\ B=\begin{bmatrix}{-4} & {2} \\ {3} & {7}\end{bmatrix} \\ C=\begin{bmatrix}{6} & {-1} & {} \\ {2} & {0} & {} \\ {-3} & {5} & {}\end{bmatrix} \end{gathered}`

To find AB, we multiply the elements of each row of matrix A by the elements of each column matrix B, and sum the products as follows:

`\begin{gathered} AB=\begin{bmatrix}{3} & {4} & {} \\ {-5} & {2} & {} \\ {1} & {0} & {}\end{bmatrix}\begin{bmatrix}{-4} & {2} \\ {3} & {7}\end{bmatrix} \\ =\begin{bmatrix}{(3*-4)+(4*3)} & {(3*2)+(4*7)} & {} \\ {(-5*-4)+(2*3)} & {(-5*2)+(2*7)} & {} \\ {(1*-4)+(0*-3)} & {(1*2)+(0*7)} & {}\end{bmatrix} \\ =\begin{bmatrix}{-12+12} & {6+28} & {} \\ {20+6} & {-10+14} & {} \\ {-4+0} & {2+0} & {}\end{bmatrix} \\ =\begin{bmatrix}{0} & {34} & {} \\ {26} & {4} & {} \\ {-4} & {2} & {}\end{bmatrix} \end{gathered}`

Hence, the product AB is

`\begin{bmatrix}{0} & {34} & {} \\ {26} & {4} & {} \\ {-4} & {2} & {}\end{bmatrix}`