# Matrix is Invertible iff Rank equals Order

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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 rank also equals $n$.

## Proof

## Sources

- 1998: Richard Kaye and Robert Wilson:
*Linear Algebra*... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.6$ Determinant and trace