Question

If $A$ is matrix of order $m\times n$ and $B$ is a matrix such that $A{B}^{\prime }$ and $B^{\prime }A$ are both defined, then order of matrix $B$ is

• A. $m\times m$
• B. $n\times n$
• C. $n\times m$
• D. $m\times n$

Answer 1

The correct option is D $m\times n$

Let $B$ is $p\times q$ matrix

$\Rightarrow B^{\prime }$ has $q\times p$ order.

In order to define $A{B}^{\prime },$ we must have

$\left\{\right(m\times n)\times (q\times p\left)\right\}\Rightarrow n=q$

Also, for $B^{\prime }A$ to be defined,

$\left\{\right(q\times p)\times (m\times n\left)\right\}\Rightarrow p=m$

$\Rightarrow B$ is $m\times n$ matrix

Hence, D will be correct answer.