Determinants and diagonalizability

Question

Does the determinant of a matrix affect if it is diagonalizable or not?

Like, if $\det(A) = 0$ does that mean the matrix is NOT diagonalizable?

Answer 1

No, and the zero matrix is an obvious example in which $\det(A) = 0$ and is itself diagonal. Also note that any diagonal matrix with at least one zero entry on the diagonal has determinant 0.

Answer 2

$A = \begin{bmatrix} 1 & 0 \\0 & 0 \end{bmatrix}$. $\det A = 0$, $A$ is diagonal.

$B = \begin{bmatrix} 0 & 1 \\0 & 0 \end{bmatrix}$. $\det B = 0$, $B$ is not diagonalizable.

$C = \begin{bmatrix} 1 & 0 \\0 & 1 \end{bmatrix}$. $\det C = 1$, $C$ is diagonal.

$D = \begin{bmatrix} 1 & 1 \\0 & 1 \end{bmatrix}$. $\det D = 1$, $D$ is not diagonalizable.