linear algebra - Proofs Invertible & Diagonal matrix

Given:
$P$ is an invertible matrix.

$D$ is a diagonal matrix.

$A$ is an $n\times n$ matrix.

AND

$A = PDP^{-1}$

Prove that the determinant of A equals the product of the diagonal entries of $D$.

Answer

Using $|ABC|=|A||B||C|$ and $|A|=\frac{1}{|A^{-1}|}$,

$\displaystyle|A|=|P D P^{-1} | =|P||D||P^{-1}|=|P||D|\frac{1}{|P|}=|D|$.

$D$ is diagonal so $|D|=\prod_{n\geq 1} D_{nn}$, that is, the determinant is equal to the product of the diagonal elements.

$|A|=\prod_{n\geq 1} D_{nn}$ as required