Given $A = \left[ \begin{array} { c c }
7 & 2 \
- 9 & 6
\end{array} \right]$ , $B = \left[ \begin{array} { l l }
3 & 5
\end{array} \right]$ , and $C = \left[ \begin{array} { c c c }
- 1 & 4 & 8 \
2 & 0 & 3
\end{array} \right]$ , which of the following can be calculated
A) $B C + A$
B) $( B A ) C$
C) $B C - A$
D) $B ( A C )$
E) Both $( B A ) C$ and $B ( A C )$

Question

Given $A = \left[ \begin{array} { c c } 
7 & 2 \
- 9 & 6
\end{array} \right]$ , $B = \left[ \begin{array} { l l } 
3 & 5
\end{array} \right]$ , and $C = \left[ \begin{array} { c c c } 
- 1 & 4 & 8 \
2 & 0 & 3
\end{array} \right]$ , which of the following can be calculated   
A) $B C + A$
B) $( B A ) C$
C) $B C - A$
D) $B ( A C )$
E) Both $( B A ) C$ and $B ( A C )$

Quizplus · Accepted Answer

Matrix multiplication is defined when the number of columns in the first matrix matches the number of rows in the second matrix. For $B A$, $B$ is $1 \times 2$ and $A$ is $2 \times 2$, so $B A$ can be calculated. For $A C$, $A$ is $2 \times 2$ and $C$ is $2 \times 3$, so $A C$ can also be calculated. Therefore, both $( B A ) C$ and $B ( A C )$ can be calculated, as the resulting matrices from $B A$ and $A C$ will have dimensions that allow further multiplication with $C$ and $B$, respectively.

Given A=[72−96]A = \left[ \begin{array} { c c } 7 & 2 \\- 9 & 6\end{array} \right]A=[7−9​26​]

Given $A = \left[ \begin{array} { c c } 7 & 2 \\- 9 & 6\end{array} \right]$