Question:
If $A$ is $3 \times 4$ matrix and $B$ is a matrix such that $A^{\prime} B$ and $B A^{\prime}$ are both defined. Then, $B$ is of the type
(a) $3 \times 4$
(b) $3 \times 3$
(c) $4 \times 4$
(d) $4 \times 3$
Solution:
(a) $3 \times 4$
The order of $A$ is $3 \times 4$. So, the order of $A$ ' is $4 \times 3$.
Now, both $A^{\prime} B$ and $B A^{\prime}$ are defined. So, the number of columns in $A^{\prime}$ should be equal to the number of rows in $B$ for $A^{\prime} B$. Also, the number of columns in $B$ should be equal to number of rows in $A$ ' for $B A$ '.
Hence, the order of matrix B is 3