# Let $A$ be a $2\times 2$ matrix with complex entries which is non-zero and non-diagonal. Pick out the cases when $A$ is diagonalizable

Let $A$ be a $2 × 2$ matrix with complex entries which is non-zero and non-diagonal. Pick out the cases when $A$ is diagonalizable.

• (a) $A^2 = I$.
• (b) $A^2 = 0$.
• (c) All eigenvalues of $A$ are equal to $2$.

(a) is true since eigen values are distinct. But what about (b) and (c)?

-
A matrix $A$ is diagonalizable if and only if its minimal polynomial factors into distinct linear factors (in the field over which the matrix is defined). Now you can check if this is the true for each case. – sos440 Sep 1 '12 at 16:53

For (b) consider $$A=\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}$$

For (C) consider $$A=\begin{pmatrix}2 & 1 \\ 0 & 2\end{pmatrix}$$

-