# Matrix is Invertible iff Determinant has Multiplicative Inverse

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## Theorem

Let $R$ be a commutative ring with unity.

Let $\mathbf A \in R^{n \times n}$ be a square matrix of order $n$.

Then $\mathbf A$ is invertible if and only if its determinant is invertible in $R$.

If this is the case, then:

- $\mathbf A^{-1} = \map \det {\mathbf A}^{-1} \cdot \adj {\mathbf A}$

where $\adj {\mathbf A}$ is the adjugate of $\mathbf A$.

If $R$ is one of the standard number fields $\Q$, $\R$ or $\C$, this translates into:

- $\mathbf A$ is invertible if and only if its determinant is non-zero.

## Proof

### Necessary Condition

Let $\mathbf A$ be invertible.

Let $1_R$ denote the unity of $R$.

Let $\mathbf I_n$ denote the unit matrix of order $n$.

Then:

\(\displaystyle 1_R\) | \(=\) | \(\displaystyle \map \det {\mathbf I_n}\) | Determinant of Unit Matrix | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \det {\mathbf A \mathbf B}\) | Definition of Inverse Matrix | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \det {\mathbf A} \, \map \det {\mathbf B}\) | Determinant of Matrix Product |

This shows that:

- $\map \det {\mathbf B} = \dfrac 1 {\map \det {\mathbf A} }$

$\Box$

### Sufficient Condition

Let $\map \det {\mathbf A}$ be invertible in $R$.

From Matrix Product with Adjugate Matrix:

\(\displaystyle \mathbf A \cdot \adj {\mathbf A}\) | \(=\) | \(\displaystyle \map \det {\mathbf A} \cdot \mathbf I_n\) | |||||||||||

\(\displaystyle \adj {\mathbf A} \cdot \mathbf A\) | \(=\) | \(\displaystyle \map \det {\mathbf A} \cdot \mathbf I_n\) |

Thus:

\(\displaystyle \mathbf A \cdot \paren {\map \det {\mathbf A}^{-1} \cdot \adj {\mathbf A} }\) | \(=\) | \(\displaystyle \mathbf I_n\) | |||||||||||

\(\displaystyle \paren {\map \det {\mathbf A}^{-1} \cdot \adj {\mathbf A} } \cdot \mathbf A\) | \(=\) | \(\displaystyle \mathbf I_n\) |

Thus $\mathbf A$ is invertible, and:

- $\mathbf A^{-1} = \map \det {\mathbf A}^{-1} \cdot \adj {\mathbf A}$

$\blacksquare$

## Also see

## Sources

- 1994: Robert Messer:
*Linear Algebra: Gateway to Mathematics*: $\S 7.4$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): $\text{A}.2$: Linear algebra and determinants: Theorem $\text{A}.9 \ (2)$ - 1980: A.J.M. Spencer:
*Continuum Mechanics*... (previous) ... (next): $2.1$: Matrices