Does diagonalizable imply invertible? Does invertible imply diagonalizable?
Solution
Neither implication holds.
A diagonalizable matrix need not be invertible. For example, \[ A=\begin{pmatrix}0&0\\0&1\end{pmatrix} \] is already diagonal, but \[ \det A=0, \] so it is not invertible.
An invertible matrix need not be diagonalizable. For example, \[ B=\begin{pmatrix}1&1\\0&1\end{pmatrix} \] has determinant \(1\), so it is invertible. But its only eigenvalue is \(1\), and \[ \ker(B-I) \] is one-dimensional, so \(B\) does not have a basis of eigenvectors and is not diagonalizable.