Question

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

• A. AB and BA are square matrices.
• B. AB and BA are symmetric matrices.
• C. AB and BA are skewsymmetric matrices.
• D. None of these.

Answer 1

Answer: (A)

AB and BA are square matrices.

If both AB and BA exist, then the number of columns of A is equal to the number of rows of B. Therefore,
$n+5=m$ ....... (1)
And the number of columns of B is equal to the number of rows of A. Therefore,
$11-n=m$ ...... (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 AB are BA are square matrices.