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

Problems and solutions in Linear Algebra

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.

 
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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.


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