If A is matrix of order m × n and B is a matrix such that AB’ and B’A are both defined, then order of matrix B is

The correct option is m × n.

Let the order of matrix $B$ be $x\times y$.
This means that matrix $B$ has $x$ rows and $y$ columns.

The transpose of matrix $B$, denoted as $B^{\prime }$, is obtained by interchanging its rows and columns.
Therefore, the order of matrix $B^{\prime }$ is $y\times x$.

We are given that the order of matrix $A$ is $m\times n$.

For the matrix product $A{B}^{\prime }$ to be defined, the number of columns in the first matrix ($A$) must be equal to the number of rows in the second matrix ($B^{\prime }$).
The number of columns in $A$ is $n$, and the number of rows in $B^{\prime }$ is $y$.
Therefore, we must have:
$y=n$

Similarly, for the matrix product $B^{\prime }A$ to be defined, the number of columns in the first matrix ($B^{\prime }$) must be equal to the number of rows in the second matrix ($A$).
The number of columns in $B^{\prime }$ is $x$, and the number of rows in $A$ is $m$.
Therefore, we must have:
$x=m$

Substituting the values of $x$ and $y$ back into the order of matrix $B$ ($x\times y$), we get the order of matrix $B$ as $m\times n$.