Is the Product of a Nilpotent Matrix and an Invertible Matrix Nilpotent?

Nilpotent Matrix Problems and Solutions

Problem 77

A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix.
Show that

(a) If $A$ is a nilpotent $n \times n$ matrix and $B$ is an $n\times n$ matrix such that $AB=BA$. Show that the product $AB$ is nilpotent.

(b) Let $P$ be an invertible $n \times n$ matrix and let $N$ be a nilpotent $n\times n$ matrix. Is the product $PN$ nilpotent? If so, prove it. If not, give a counterexample.

 

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Hint.

For (b), the statement is false. Try to find a counter example.
A typical nilpotent matrix is an upper triangular matrix whose diagonal entries are all zero.

Proof.

(a) Show that $AB$ is nilpotent

Since $A$ is nilpotent, there exists a positive integer $k$ such that $A^k=O$. Then we have
\[(AB)^k=(AB)(AB)\cdots (AB)=A^kB^k=OB^k=O.\] Here in the second equality, we used the assumption that $AB=BA$.
Thus we have $(AB)^k=O$, hence the product matrix $AB$ is nilpotent.

 

(b) Is $PN$ nilpotent?

In general, the product $PN$ of an invertible matrix $P$ and a nilpotent matrix $N$ is not nilpotent.
Here is a counterexample.
Let
\[P=\begin{bmatrix}
1 & 0 & 0 \\
1 &1 &0 \\
0 & 0 & 1
\end{bmatrix} \text{ and }
N=\begin{bmatrix}
0 & 1 & 1 \\
0 &0 &1 \\
0 & 0 & 0
\end{bmatrix}.\] Then the matrix $P$ is invertible since $\det(P)=1$.
(Note that $P$ is a lower triangular matrix. So the determinant is the product of diagonal entries.)
Also, it is easy to see by direct computation that $N^3=O$, hence $N$ is nilpotent. Indeed,
\[N^2=\begin{bmatrix}
0 & 0 & 1 \\
0 &0 &0 \\
0 & 0 & 0
\end{bmatrix} \] and
\[
N^3=N^2N=\begin{bmatrix}
0 & 0 & 1 \\
0 &0 &0 \\
0 & 0 & 0
\end{bmatrix}
\begin{bmatrix}
0 & 1 & 1 \\
0 &0 &1 \\
0 & 0 & 0
\end{bmatrix}=O.\]

Now the product $PN$ is
\[PN=\begin{bmatrix}
0 & 1 & 1 \\
0 &1 &2 \\
0 & 0 & 0
\end{bmatrix}.\] We show that $PN$ is not nilpotent.
We have
\[(PN)^2=\begin{bmatrix}
0 & 1 & 2 \\
0 &1 &2 \\
0 & 0 & 0
\end{bmatrix}\] \[(PN)^3=(PN)^2(PN)=\begin{bmatrix}
0 & 1 & 2 \\
0 &1 &2 \\
0 & 0 & 0
\end{bmatrix}\begin{bmatrix}
0 & 1 & 1 \\
0 &1 &2 \\
0 & 0 & 0
\end{bmatrix}
=\begin{bmatrix}
0 & 1 & 2 \\
0 &1 &2 \\
0 & 0 & 0
\end{bmatrix}.\] This calculation shows that
\[(PN)^k=\begin{bmatrix}
0 & 1 & 2 \\
0 &1 &2 \\
0 & 0 & 0
\end{bmatrix}\neq O \text{ for all } k \geq 2.\] Thus $PN$ is not nilpotent. In conclusion, the product $PN$ of the invertible matrix $P$ and the nilpotent matrix $N$ is not nilpotent.


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