Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$.

Nilpotent Matrix Problems and Solutions

Problem 305

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.)

 
LoadingAdd to solve later

Sponsored Links


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 $3$.


LoadingAdd to solve later

Sponsored Links

More from my site

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Introduction to Linear Algebra at the Ohio State University quiz problems and solutions
Quiz 5: Example and Non-Example of Subspaces in 3-Dimensional Space

Problem 1 Let $W$ be the subset of the $3$-dimensional vector space $\R^3$ defined by \[W=\left\{ \mathbf{x}=\begin{bmatrix} x_1 \\ x_2...

Close