Question

If AB = 0 for two 2 x 2 matrices A and B, then BA must be zero matrix as well.

Short answer

The above statement is false.

If AB = 0 for two 2 x 2 matrices A and B, then BA may not be a zero matrix.

Step-by-step solution

Taking an example

Let $A=\left[\begin{array}{cc}0& -1\\ 0& 2\end{array}\right]\text{andB=}\left[\begin{array}{cc}3& 5\\ 0& 0\end{array}\right]$are two 2x2 matrices such that

$AB=\left[\begin{array}{cc}0& -1\\ 0& 2\end{array}\right]\left[\begin{array}{cc}3& 5\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$\Rightarrow AB=0$

Now,

$BA=\left[\begin{array}{cc}3& 5\\ 0& 0\end{array}\right]\left[\begin{array}{cc}0& -1\\ 0& 2\end{array}\right]=\left[\begin{array}{cc}0& 7\\ 0& 0\end{array}\right]$

$\Rightarrow BA\ne 0$

Final Answer

Since we have find two 2 X 2 matrices$A=\left[\begin{array}{cc}0& -1\\ 0& 2\end{array}\right]\text{andB=}\left[\begin{array}{cc}3& 5\\ 0& 0\end{array}\right]$ such that

$AB=0\text{but BA}\ne \text{0}$, hence the given statement is false.

Thus, if AB = 0 for two 2 x 2 matrices A and B, then BA may not be a zero matrix.