ThmDex – An index of mathematical definitions, results, and conjectures.
Determinant of a diagonal complex matrix with constant diagonal

Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix such that
 (i) $\lambda \in \mathbb{C}$ is a D1207: Complex number (i) $$A = \begin{bmatrix} \lambda & 0 & \cdots & 0 \\ 0 & \lambda & \vdots & 0 \\ \vdots & \cdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda \end{bmatrix}$$
Then $$\text{Det} A = \lambda^N$$

Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix such that
 (i) $\lambda \in \mathbb{C}$ is a D1207: Complex number (ii) $I_N \in \mathbb{C}^{N \times N}$ is a D5699: Complex identity matrix (iii) $$A = \lambda I_N$$
Then $$\text{Det} A = \lambda^N$$
Proofs
Proof 0
Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix such that
 (i) $\lambda \in \mathbb{C}$ is a D1207: Complex number (i) $$A = \begin{bmatrix} \lambda & 0 & \cdots & 0 \\ 0 & \lambda & \vdots & 0 \\ \vdots & \cdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda \end{bmatrix}$$
This result is a particular case of R5069: Determinant of a diagonal complex matrix. $\square$