ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4107 on D763: Matrix determinant
Subresult of R4106: Determinant of real diagonal matrix

Determinant of real diagonal matrix with constant diagonal

Formulation 0

Let $A \in \mathbb{R}^{N \times N}$ be a D6160: Real square matrix such that

(i)	$\lambda \in \mathbb{R}$ is a D993: Real number
(ii)	\begin{equation} A = \begin{bmatrix} \lambda & 0 & \cdots & 0 \\ 0 & \lambda & \vdots & 0 \\ \vdots & \cdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda \end{bmatrix} \end{equation}

Then \begin{equation} \text{Det} A = \lambda^N \end{equation}

Formulation 1

Let $A \in \mathbb{R}^{N \times N}$ be a D6160: Real square matrix such that

(i)	$\lambda \in \mathbb{R}$ is a D993: Real number
(ii)	$I_N \in \mathbb{R}^{N \times N}$ is a D5621: Real identity matrix
(iii)	$A = \lambda I_N$

Then \begin{equation} \text{Det} A = \lambda^N \end{equation}

Proofs

Proof 0

Let $A \in \mathbb{R}^{N \times N}$ be a D6160: Real square matrix such that

(i)	$\lambda \in \mathbb{R}$ is a D993: Real number
(ii)	\begin{equation} A = \begin{bmatrix} \lambda & 0 & \cdots & 0 \\ 0 & \lambda & \vdots & 0 \\ \vdots & \cdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda \end{bmatrix} \end{equation}

This result is a particular case of R4106: Determinant of real diagonal matrix. $\square$