# If the Matrix Product $AB=0$, then is $BA=0$ as Well?

## Problem 98

Let $A$ and $B$ be $n\times n$ matrices. Suppose that the matrix product $AB=O$, where $O$ is the $n\times n$ zero matrix.

Is it true that the matrix product with opposite order $BA$ is also the zero matrix?

If so, give a proof. If not, give a counterexample.

## Solution.

The statement is in general not true. We give a counter example.

Consider the following $2\times 2$ matrices.

\[A=\begin{bmatrix}

0 & 1\\

0& 1

\end{bmatrix} \text{ and } \begin{bmatrix}

1 & 1\\

0& 0

\end{bmatrix}.\]

Then we compute

\[AB=\begin{bmatrix}

0 & 1\\

0& 1

\end{bmatrix}

\begin{bmatrix}

1 & 1\\

0& 0

\end{bmatrix}

=\begin{bmatrix}

0 & 0\\

0& 0

\end{bmatrix}=O.\]
Thus the matrix product $AB$ is the $2\times 2$ zero matrix $O$.

On the other hand, we compute

\[BA=\begin{bmatrix}

1 & 1\\

0& 0

\end{bmatrix}

\begin{bmatrix}

0 & 1\\

0& 1

\end{bmatrix}=\begin{bmatrix}

0 & 2\\

0& 0

\end{bmatrix}.\]

Thus the matrix product $BA$ is not the zero matrix.

Therefore the statement is not true in general.

Add to solve later