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.

If matrix product $AB$ is a square, then is $BA$ a square matrix?
Let $A$ and $B$ are matrices such that the matrix product $AB$ is defined and $AB$ is a square matrix.
Is it true that the matrix product $BA$ is also defined and $BA$ is a square matrix? If it is true, then prove it. If not, find a […]

Symmetric Matrices and the Product of Two Matrices
Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings.
(a) The product $AB$ is symmetric if and only if $AB=BA$.
(b) If the product $AB$ is a diagonal matrix, then $AB=BA$.
Hint.
A matrix $A$ is called symmetric if $A=A^{\trans}$.
In […]

Basis For Subspace Consisting of Matrices Commute With a Given Diagonal Matrix
Let $V$ be the vector space of all $3\times 3$ real matrices.
Let $A$ be the matrix given below and we define
\[W=\{M\in V \mid AM=MA\}.\]
That is, $W$ consists of matrices that commute with $A$.
Then $W$ is a subspace of $V$.
Determine which matrices are in the subspace $W$ […]

10 True or False Problems about Basic Matrix Operations
Test your understanding of basic properties of matrix operations.
There are 10 True or False Quiz Problems.
These 10 problems are very common and essential.
So make sure to understand these and don't lose a point if any of these is your exam problems.
(These are actual exam […]

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 $(A-B)(A+B)=A^2-B^2$.
Hint.
In general, matrix multiplication is not commutative: $AB$ and $BA$ might be different.
Solution.
Let us calculate $(A-B)(A+B)$ as […]

Linear Properties of Matrix Multiplication and the Null Space of a Matrix
Let $A$ be an $m \times n$ matrix.
Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in \calN(A)$ and $\mathbf{v} \in \calN(A)$.
Let $\mathbf{w}=3\mathbf{u}-5\mathbf{v}$.
Then find $A\mathbf{w}$.
Hint.
Recall that the null space of an […]

Is the Determinant of a Matrix Additive?
Let $A$ and $B$ be $n\times n$ matrices, where $n$ is an integer greater than $1$.
Is it true that
\[\det(A+B)=\det(A)+\det(B)?\]
If so, then give a proof. If not, then give a counterexample.
Solution.
We claim that the statement is false.
As a counterexample, […]