Diagonalizable matrix $A$ invertible also?

If a matrix $A$ is diagonalizable, is $A$ invertible?

I know that $P^{-1}AP = \text{some diagonal matrix}$ and therefore $P$ is invertible, but not sure of $A$ itself.

Solution 1:

If that diagonal matrix has any zeroes on the diagonal, then $A$ is not invertible. Otherwise, $A$ is invertible. The determinant of the diagonal matrix is simply the product of the diagonal elements, but it's also equal to the determinant of $A$.

Solution 2:

No. For instance, the zero matrix is diagonalizable, but isn't invertible.

Solution 3:

A square matrix is invertible if an only if its kernel is $0$, and an element of the kernel is the same thing as an eigenvector with eigenvalue $0$, since it is mapped to $0$ times itself, which is $0$.

When we diagonalize a matrix, we pick a basis so that the matrix's eigenvalues are on the diagonal, and all other entries are $0$. So if $P^{-1}AP$ is diagonal, then $P^{-1}AP$ is invertible if an only if none of its diagonal entries (eigenvalues) are $0$.

$P^{-1}AP$ is invertible if an only if $A$ is invertible because they are the same transformation, written with different bases. Alternatively, note that $(P^{-1}AP)^{-1}=P^{-1}A^{-1}P$.