Question

Matrix $A$ has $m$ rows and $n+5$ columns; matrix $B$ has $m$ rows and $11-n$ columns. If both $A B$ and $B A$ exist, then

• A. $A B$ and $B A$ are square matrix
• B. $A B$ and $B A$ are of order $8 \times 8$ and $3 \times 13$, respectively
• C. $A B=B A$
• D. None of these

Answer 1

Answer: (A)

If both $A B$ and $B A$ exists then number of columns of $A$ = number of rows of $B$
$\Rightarrow n+5=m$ ...(1)
and number of columns of $B=$ number of row of $A$
$\Rightarrow 11-n=m$ ...(2)
Solving (1) and (2) we get $n=3$ and $m=8$
Hence, $A$ has order $8 \times 8$ and $B$ has order $8 \times 8$
Hence, both $A B$ and $B A$ are square matrix