# Determinant of Unit Matrix

## Theorem

Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

The determinant of the unit matrix of order $n$ over $R$ is equal to $1_R$.

## Proof

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

The proof proceeds by induction.

For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:

- $\map \det {\mathbf I_n} = 1_R$

By definition of Determinant of Order $1$:

- $\begin {vmatrix} a_{1 1} \end {vmatrix} = a_{1 1}$

In this case $a_{1 1} = 1_R$.

Thus $\map P 1$ is seen to hold.

### Basis for the Induction

\(\ds \map \det {\mathbf I_2}\) | \(=\) | \(\ds \begin {vmatrix} 1_R & 0_R \\ 0_R & 1_R \end {vmatrix}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 1_R \cdot 1_R - 0_R \cdot 0_R\) | Definition of Determinant of Order 2 | |||||||||||

\(\ds \) | \(=\) | \(\ds 1_R\) |

Thus $\map P 2$ is seen to hold.

This is the basis for the induction.

### Induction Hypothesis

Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:

- $\map \det {\mathbf I_k} = 1_R$

from which it is to be shown that:

- $\map \det {\mathbf I_{k + 1} } = 1_R$

### Induction Step

This is the induction step:

\(\ds \mathbf I_{k + 1}\) | \(=\) | \(\ds \begin {bmatrix} 1_R & 0_R \\ 0_R & \mathbf I_n \end {bmatrix}\) | Definition of Unit Matrix | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \map \det {\mathbf I_{k + 1} }\) | \(=\) | \(\ds \begin {vmatrix} 1_R & 0_R \\ 0_R & \mathbf I_n \end {vmatrix}\) | Definition of Determinant of Matrix | ||||||||||

\(\ds \) | \(=\) | \(\ds \map \det {\mathbf I_k}\) | Determinant with Unit Element in Otherwise Zero Row | |||||||||||

\(\ds \) | \(=\) | \(\ds 1_R\) | Induction Hypothesis |

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:

- $\forall n \in \Z_{\ge 0}: \map \det {\mathbf I_n} = 1_R$

$\blacksquare$

## 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: Fact $1.7 \ \text {(d)}$