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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Let $A$ and $B$ be $n\times n$ matrices. Suppose that the matrix product $AB=O$, where $O$ is the $n\times n$ zero matrix.
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Let $A$ be an $n \times n$ matrix such that $\tr(A^n)=0$ for all $n \in \N$.
Then prove that $A$ is a nilpotent matrix. Namely there exist a positive integer $m$ such that $A^m$ is the zero matrix.
Steps.
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Let $A_1, A_2, \dots, A_m$ be $n\times n$ Hermitian matrices. Show that if
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Hint.
Recall that a complex matrix $A$ is Hermitian if […]
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Definition.
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\[A=B+iC,\]
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\end{bmatrix}\]
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