Solution:
Given that matrices A, B, C as follows:
[tex]\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}[/tex]
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:
[tex]\begin{gathered} AB=\begin{bmatrix}{3} & {4} & {} \\ {-5} & {2} & {} \\ {1} & {0} & {}\end{bmatrix}\begin{bmatrix}{-4} & {2} \\ {3} & {7}\end{bmatrix} \\ =\begin{bmatrix}{(3\times-4)+(4\times3)} & {(3\times2)+(4\times7)} & {} \\ {(-5\times-4)+(2\times3)} & {(-5\times2)+(2\times7)} & {} \\ {(1\times-4)+(0\times-3)} & {(1\times2)+(0\times7)} & {}\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}[/tex]
Hence, the product AB is
[tex]\begin{bmatrix}{0} & {34} & {} \\ {26} & {4} & {} \\ {-4} & {2} & {}\end{bmatrix}[/tex]
