Let $A$ be an $n\times n$ singular matrix.
Then prove that there exists a nonzero $n\times n$ matrix $B$ such that
\[AB=O,\]
where $O$ is the $n\times n$ zero matrix.

Recall that an $n \times n$ matrix $A$ is called singular if the equation
\[A\mathbf{x}=\mathbf{0}\]
has a nonzero solution $\mathbf{x}\in \R^n$.

Proof.

Since $A$ is singular, there exists a nonzero vector $\mathbf{b} \in \R^n$ such that
\[A\mathbf{b}=\mathbf{0}.\]

Then we define the $n\times n$ matrix $B$ whose first column is the vector $\mathbf{b}$ and the other entries are zero. That is
\[B=\begin{bmatrix}
\mathbf{b} & \mathbf{0} & \cdots & \mathbf{0}
\end{bmatrix}.\]
Since the vector $\mathbf{b}$ is nonzero, the matrix $B$ is nonzero.

With this choice of $B$, we have
\begin{align*}
AB&=A\begin{bmatrix}
\mathbf{b} & \mathbf{0} & \cdots & \mathbf{0}
\end{bmatrix} \\
&=\begin{bmatrix}
A\mathbf{b} & A\mathbf{0} & \cdots & A\mathbf{0}
\end{bmatrix}\\
&=\begin{bmatrix}
\mathbf{0} & \mathbf{0} & \cdots & \mathbf{0}
\end{bmatrix}
=O.
\end{align*}
Hence we have proved that $AB=O$ with the nonzero matrix $B$.

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

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 […]

Two Matrices are Nonsingular if and only if the Product is Nonsingular
An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$.
Using the definition of a nonsingular matrix, prove the following statements.
(a) If $A$ and $B$ are $n\times […]

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 […]

Sum of Squares of Hermitian Matrices is Zero, then Hermitian Matrices Are All Zero
Let $A_1, A_2, \dots, A_m$ be $n\times n$ Hermitian matrices. Show that if
\[A_1^2+A_2^2+\cdots+A_m^2=\calO,\]
where $\calO$ is the $n \times n$ zero matrix, then we have $A_i=\calO$ for each $i=1,2, \dots, m$.
Hint.
Recall that a complex matrix $A$ is Hermitian if […]

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 […]