Diagonal matrix/No homothety/Does not commute/Exercise

Let

${}D={\begin{pmatrix}a_{11}&0&\cdots &\cdots &0\\0&a_{22}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&a_{n-1\,n-1}&0\\0&\cdots &\cdots &0&a_{nn}\end{pmatrix}}\,.$

Show that ${}D$ commutes with all ${}n\times n$-matrices ${}M$ if and only if all its diagonal entries coincide.

Create a solution