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

Let $A = {\left[ {{a_{ij}}} \right]_{m \times n}}$ and $B = {\left[ {{b_{ij}}} \right]_{p \times q}}$

$\therefore$ ${B^\prime } = {\left[ {{b_{jj}}} \right]_{q \times p}}$
Now, $A{B^\prime }$ is defined, so $n = q$

and ${B^\prime }A$ is also defined, so $p = m$

$\therefore$ Order of ${B^\prime } = {\left[ {{b_{ji}}} \right]_{n \times m}}$

and order of $B = {\left[ {{b_{ij}}} \right]_{m \times n}}$