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Linear Algebra

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Moderate

Diagonal Matrices

LINALG-WECNJH

Let $A$ be a $3\times3$ diagonal matrix.

Which one of the following statements is necessarily TRUE?

A

$A$ commutes with any $3\times3$ symmetric matrix

B

$A$ is invertible

C

$A$ commutes with the matrix $\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix}$

D

$A$ commutes with the standard matrix of any rotation of the plane spanned by the first two standard basis vectors of $\mathbb{R}^3$

E

$A$ commutes with any other diagonal $3\times3$ matrix