# 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.

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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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##### True or False: $(A-B)(A+B)=A^2-B^2$ for Matrices $A$ and $B$

Let $A$ and $B$ be $2\times 2$ matrices. Prove or find a counterexample for the statement that $(A-B)(A+B)=A^2-B^2$.

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