Question

Matrix $A$ has $m$ rows and$(n+5)$ columns, matrix $B$ has $m$ rows and $(11-n)$ column. If both $AB$ ans $AB$ exist, then

• A. $AB$ and $BA$ are square matrix
• B. $AB$ and$BA$ are of orders $8\times 8$ and $3\times 13$ respectively
• C. $AB=BA$
• D. $Noneoftheabove$

Answer 1

The correct option is A $AB$ and $BA$ are square matrix

If both AB and BA exist,

then the number of columns of A is equal to the number of rows of B.

Therefore,

$n+5=m-\left(1\right)$

And the number of columns of B is equal to the number of rows of A. Therefore,

$11-n=m-\left(2\right)$

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.