Find two different \(2 \times 2\) matrices \(A\) such that \(A^2 = 0\) but \(A \neq 0\).
Solution
Two examples are \[ A_1=\begin{bmatrix}0&1\\0&0\end{bmatrix}, \qquad A_2=\begin{bmatrix}0&0\\1&0\end{bmatrix}. \] Both are nonzero, and direct multiplication gives \[A_1^2=A_2^2=0.\] Since \(A_1\ne A_2\), these are two different required matrices.