In the basis that diagonalizes the matrix, the matrix can be written :

Where are the eigenvalues of the matrix.

The determinant in this base is obviously the product of the

So if one of the eigenvalues is 0, the determinant is 0.

And since the determinant doesn't change with the basis, we have proved that det A = 0, and hence it is not invertible.

Edit : waaaah 39th post