Question

If $A$ is a matrix of order $m\times n$ and $B$ is a matrix such that $A{B}^{\prime }$ and $B^{\prime }A$ are both defined, the 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 $B=[b_{ij}{]}_{p\times q}$
As $A{B}^{\prime }$ is defined so we get, order of $A{B}^{\prime }$ as $(m\times n)(q\times p)\Rightarrow q=n$
Also, $B^{\prime }A$ is defined so we get, order of $B^{\prime }A$ as $(q\times p)(m\times n)\Rightarrow p=m$

Therefore, order of matrix $B$ is $m\times n$