Find a nonzero $3\times 3$ matrix $A$ such that $A^2\neq O$ and $A^3=O$, where $O$ is the $3\times 3$ zero matrix.

(Such a matrix is an example of a nilpotent matrix. See the comment after the solution.)

Solution.

For example, let $A$ be the following $3\times 3$ matrix.
\[A=\begin{bmatrix}
0 & 1 & 0 \\
0 &0 &1 \\
0 & 0 & 0
\end{bmatrix}.\] Then $A$ is a nonzero matrix and we have
\[A^2=\begin{bmatrix}
0 & 1 & 0 \\
0 &0 &1 \\
0 & 0 & 0
\end{bmatrix}\begin{bmatrix}
0 & 1 & 0 \\
0 &0 &1 \\
0 & 0 & 0
\end{bmatrix}
=\begin{bmatrix}
0 & 0 & 1 \\
0 &0 &0 \\
0 & 0 & 0
\end{bmatrix}\neq O.\]

The third power of $A$ is
\[A^3=A^2A=\begin{bmatrix}
0 & 0 & 1 \\
0 &0 &0 \\
0 & 0 & 0
\end{bmatrix}\begin{bmatrix}
0 & 1 & 0 \\
0 &0 &1 \\
0 & 0 & 0
\end{bmatrix}=
\begin{bmatrix}
0 & 0 & 0 \\
0 &0 &0 \\
0 & 0 & 0
\end{bmatrix}=O.\] Thus, the nonzero matrix $A$ satisfies the required conditions $A^2\neq O, A^3=O$.

Comment.

A square matrix $A$ is called nilpotent if there is a non-negative integer $k$ such that $A^k$ is the zero matrix.
The smallest such an integer $k$ is called degree or index of $A$.

The matrix $A$ in the solution above gives an example of a $3\times 3$ nilpotent matrix of degree $3$.

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