Question

State whether the following statement is True or False:
If $A$ and $B$ are two square matrices of the same order, then $AB=BA$.

• A. True
• B. False

Answer 1

The correct option is B False

Let $A=\left[\begin{array}{cc}1& 0\\ 2& 3\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right]$ Hence,
$AB=\left[\begin{array}{cc}1\times 1+0\times 1& 1\times 0+0\times 0\\ 2\times 1+3\times 1& 2\times 0+3\times 0\end{array}\right]$
$\Rightarrow AB=\left[\begin{array}{cc}1& 0\\ 5& 0\end{array}\right]$
And $BA=\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right]\left[\begin{array}{cc}1& 0\\ 2& 3\end{array}\right]$
$=\left[\begin{array}{cc}1\times 1+0\times 2& 1\times 0+0\times 3\\ 1\times 1+0\times 2& 1\times 0+0\times 3\end{array}\right]$
$\Rightarrow BA=\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right]$

From eq.$\left(1\right)$ and eq.$\left(2\right)$
$AB\ne BA$
Hence, the given statement is false.