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

## 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.

**(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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