$R(A) = C(A)$ for non-invertible matrix implies diagonalizable?

Question

Let $A$ be a square non-invertible matrix such that $Rows(A) = Cols(A)$ (They span the same vector space).

Prove or disprove - $A$ is diagonalizable.

I tried very hard to find a disproval example of this matrix $A$ but I always end up with a matrix the fullfil the conditions.

It feels a little bit that I might prove that such matrix $A$ is symmetric, and therefore depending on the spectral theorem it is also diagonalizable.

Any hints ? Thank you !

Answer 1

The answer is no. For instance, the matrix $$ \pmatrix{1&1&0\\0&1&0\\0&0&0} $$ has the same row and column spaces, but fails to be diagonalizable.