Question

Hello! I need a practice question explained and answered. I’m having trouble with this. - Rose

Answer 1

Evaluate the product of matrix A and matrix B to obtain the matrix AB.

`\begin{gathered} AB=\begin{bmatrix}{7} & {1} & \\ {1} & {5} & {} \\ {} & {} & {}\end{bmatrix}\begin{bmatrix}{1} & -2{\square} & 5{\square} \\ {4} & {6} & {2} \\ {} & {} & {}\end{bmatrix} \\ =\begin{bmatrix}{7+4} & {-14+6} & {35+2} \\ {1+20} & {-2+30} & {5+10} \\ {} & & {}\end{bmatrix} \\ =\begin{bmatrix}{11} & {-8} & {37} \\ {21} & {28} & {15} \\ {} & {} & {}\end{bmatrix} \end{gathered}`

Evalaute the product of AB matrix with C matrix to obtain matrix for (AB)C.

`\begin{gathered} (AB)C=\begin{bmatrix}{11} & {-8} & {37} \\ {21} & {28} & {15} \\ {} & {} & {}\end{bmatrix}\begin{bmatrix}{1} & {} & {} \\ {-2} & {} & \\ {0} & {} & {}\end{bmatrix} \\ =\begin{bmatrix}{11+16+0} & {} & {} \\ {21-56+0} & & {} \\ {} & {} & {}\end{bmatrix} \\ =\begin{bmatrix}{27} & {} & {} \\ {-35} & & {} \\ {} & {} & {}\end{bmatrix} \end{gathered}`

Evalaute the product of matrix B and matric C to obtain the matrix BC,

`\begin{gathered} BC=\begin{bmatrix}{1} & -2{\square} & 5{\square} \\ {4} & {6} & {2} \\ {} & {} & {}\end{bmatrix}\begin{bmatrix}{1} & {} & {} \\ {-2} & {} & \\ {0} & {} & {}\end{bmatrix} \\ =\begin{bmatrix}{1+4+0} & {} & {} \\ {4-12+0} & & {} \\ {} & {} & {}\end{bmatrix} \\ =\begin{bmatrix}{5} & {} & {} \\ {-8} & & {} \\ {} & {} & {}\end{bmatrix} \end{gathered}`

Evaluate the product of matrix A with matrix (BC) to obtain the matrix A(BC).

`\begin{gathered} A(BC)=\begin{bmatrix}{7} & {1} & \\ {1} & {5} & {} \\ {} & {} & {}\end{bmatrix}\begin{bmatrix}{5} & {} & {} \\ {-8} & & {} \\ {} & {} & {}\end{bmatrix} \\ =\begin{bmatrix}{35-8} & {} & {} \\ {5-40} & & {} \\ {} & {} & {}\end{bmatrix} \\ =\begin{bmatrix}{27} & {} & {} \\ {-35} & & {} \\ {} & {} & {}\end{bmatrix} \end{gathered}`

So it can be observed that A(BC) = (AB)C.