Question

State whether the following statement is True or False:
If ${\left(AB\right)}^{\prime }=B^{\prime }{A}^{\prime }$, where $A$ and $B$ are not square matrices, then number of rows in $A$ is equal to number of columns in $B$ and number of columns in $A$ is equal to number of rows in $B$.

• A. False
• B. True

Answer 1

The correct option is A False

Let $A$ is of order $m\times n$ $(m\ne n)$
and $B$ is of order $p\times q$ $(p\ne q)$
Given, ${\left(AB\right)}^{\prime }=B^{\prime }{A}^{\prime }$
$\left(A_{(m\times n)}{B}_{(p\times q)}\right)$ is defined only when $n=p$
$\therefore \text{Order of}AB\to m\times q$
$\Rightarrow \text{Order of}{\left(AB\right)}^{\prime }\to q\times m$
Now, $B^{\prime }$ is of order $q\times p$
and $A^{\prime }$ is of order $n\times m$.
We know, $p=n$
Order of $B^{\prime }{A}^{\prime }\to q\times m$
${\left(AB\right)}^{\prime }=B^{\prime }{A}^{\prime }$ is true when $n=p$
i.e., number of columns in $A$ is equal to number of rows in $B$. But it is not necessary that number of rows in matrix $A$ is same as number of columns in matrix $B$.
Hence, the given statement is false.