Question:
If $A=\left[\begin{array}{ccc}2 & -1 & 3 \\ -4 & 5 & 1\end{array}\right]$ and $B=\left[\begin{array}{cc}2 & 3 \\ 4 & -2 \\ 1 & 5\end{array}\right]$, then
(a) only $A B$ is defined
(b) only $B A$ is defined
(c) $A B$ and $B A$ both are defined
(d) $A B$ and $B A$ both are not defined
Solution:
Given: $A=\left[\begin{array}{ccc}2 & -1 & 3 \\ -4 & 5 & 1\end{array}\right]$ and $B=\left[\begin{array}{cc}2 & 3 \\ 4 & -2 \\ 1 & 5\end{array}\right]$
Order of $A$ is $2 \times 3$ and order of $B$ is $3 \times 2$.
Therefore, $A B$ and $B A$ both are defined.
Hence, the correct option is (c).