Linear Algebra

If a Matrix is the Product of Two Matrices, is it Invertible?

Problem 393

(a) Let $A$ be a $6\times 6$ matrix and suppose that $A$ can be written as
\[A=BC,\] where $B$ is a $6\times 5$ matrix and $C$ is a $5\times 6$ matrix.

Prove that the matrix $A$ cannot be invertible.

(b) Let $A$ be a $2\times 2$ matrix and suppose that $A$ can be written as
\[A=BC,\] where $B$ is a $ 2\times 3$ matrix and $C$ is a $3\times 2$ matrix.

Can the matrix $A$ be invertible?

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Solution.

(a) Prove that the matrix $A$ cannot be invertible.

Since $C$ is a $5 \times 6$ matrix, the equation
\[C\mathbf{x}=\mathbf{0}\] has a nonzero solution $\mathbf{x}_1$.
(There are more variables than equations in the system $C\mathbf{x}=\mathbf{0}$.)

It follows that we have
\begin{align*}
A\mathbf{x}_1=BC\mathbf{x}_1=B\mathbf{0}=\mathbf{0}.
\end{align*}
Since the vector $\mathbf{x}_1$ is nonzero, the matrix $A$ is a singular matrix, hence $A$ is not invertible.

(b) Can the matrix $A$ be invertible?

The answer is yes. For example consider the following $2\times 3$ matrix $B$ and $3 \times 2$ matrix $C$:
\[B=\begin{bmatrix}
1 & 0 & 0 \\
0 &1 &0
\end{bmatrix}, \qquad C=\begin{bmatrix}
1 & 0 \\
0 & 1 \\
0 &0
\end{bmatrix}.\] Then we have
\begin{align*}
A=BC=\begin{bmatrix}
1 & 0 & 0 \\
0 &1 &0
\end{bmatrix}
\begin{bmatrix}
1 & 0 \\
0 & 1 \\
0 &0
\end{bmatrix}
=
\begin{bmatrix}
1 & 0\\
0& 1
\end{bmatrix},
\end{align*}
and the matrix $A$ is invertible.

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