## A Square Root Matrix of a Symmetric Matrix

## Problem 59

Answer the following two questions with justification.

**(a)** Does there exist a $2 \times 2$ matrix $A$ with $A^3=O$ but $A^2 \neq O$? Here $O$ denotes the $2 \times 2$ zero matrix.

**(b) **Does there exist a $3 \times 3$ real matrix $B$ such that $B^2=A$ where

\[A=\begin{bmatrix}

1 & -1 & 0 \\

-1 &2 &-1 \\

0 & -1 & 1

\end{bmatrix}\,\,\,\,?\]

(*Princeton University Linear Algebra Exam*)