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

Question

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)?

Answer 1

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

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