Question

State whether the following statement is True or False:
If $A$ and $B$ are any two matrices of the same order, then ${\left(AB\right)}^{\prime }=A^{\prime }{B}^{\prime }$.

• A. True
• B. False

Answer 1

The correct option is B False

Take any two matrices $A$ and $B$ and identify ${\left(AB\right)}^{\prime }$.
Let $A=\left[\begin{array}{cc}1& 0\\ 2& 3\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right]$
Then
$AB=\left[\begin{array}{cc}1\times 1+0\times 1& 1\times 0+1\times 0\\ 2\times 1+3\times 1& 2\times 0+3\times 0\end{array}\right]$
$\Rightarrow AB=\left[\begin{array}{cc}1& 0\\ 5& 0\end{array}\right]$
Hence, ${\left(AB\right)}^{\prime }=\left[\begin{array}{cc}1& 5\\ 0& 0\end{array}\right]$ ...(1)
Now,
$A^{\prime }=\left[\begin{array}{cc}1& 2\\ 0& 3\end{array}\right]$ and $B^{\prime }=\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$
$\Rightarrow A^{\prime }{B}^{\prime }=\left[\begin{array}{cc}1& 2\\ 0& 3\end{array}\right]\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$
$\Rightarrow A^{\prime }{B}^{\prime }=\left[\begin{array}{cc}1\times 1+2\times 0& 1\times 1+2\times 0\\ 0\times 1+3\times 0& 0\times 1+3\times 0\end{array}\right]$
$\Rightarrow A^{\prime }{B}^{\prime }=\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$ ...(2)

From equation $\left(1\right)$ and equation $\left(2\right)$
${\left(AB\right)}^{\prime }\ne A^{\prime }{B}^{\prime }$
Hence, the given statement is false.