Product of Matrices is Invertible iff Matrices are Invertible

Theorem

Let $\mathbf A$ and $\mathbf B$ be square matrices of order $n$.

Let $\mathbf A \mathbf B$ denote the matrix product of $\mathbf A$ and $\mathbf B$.

Let $\mathbf I$ be the $n \times n$ unit matrix.

Then:

$\mathbf A \mathbf B$ is invertible

if and only if

both $\mathbf A$ and $\mathbf B$ are invertible.

Proof

Necessary Condition

Let both $\mathbf A$ and $\mathbf B$ be invertible.

By Matrix is Invertible iff Determinant has Multiplicative Inverse:

$\map \det {\mathbf A} \ne 0$ and $\map \det {\mathbf B} \ne 0$

where $\map \det {\mathbf A}$ denotes the determinant of $\mathbf A$.

By Determinant of Matrix Product:

$\map \det {\mathbf A} \map \det {\mathbf B} = \map \det {\mathbf A \mathbf B}$

Thus as both $\map \det {\mathbf A} \ne 0$ and $\map \det {\mathbf B} \ne 0$, it follows that:

$\map \det {\mathbf A \mathbf B} \ne 0$

Hence by Matrix is Invertible iff Determinant has Multiplicative Inverse, $\mathbf A \mathbf B$ is invertible.

Sufficient Condition

Let $\mathbf A \mathbf B$ be invertible.

Aiming for a contradiction, suppose it is not the case that both $\mathbf A$ and $\mathbf B$ are invertible.

Then by Matrix is Invertible iff Determinant has Multiplicative Inverse, either:

$\map \det {\mathbf A} = 0$

or:

$\map \det {\mathbf B} = 0$

By Determinant of Matrix Product:

$\map \det {\mathbf A} \map \det {\mathbf B} = \map \det {\mathbf A \mathbf B}$

and so:

$\map \det {\mathbf A \mathbf B} = 0$

Hence by Matrix is Invertible iff Determinant has Multiplicative Inverse, $\mathbf A \mathbf B$ is not invertible.

This contradicts the assumption that $\mathbf A \mathbf B$ is invertible.

Hence by Proof by Contradiction it follows that both $\mathbf A$ and $\mathbf B$ are invertible.

$\blacksquare$

Also see

• Inverse of Matrix Product

Sources

• 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.3$ The inverse of a matrix: Proposition $1.2$