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.

(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.

 

LoadingAdd to solve later

 

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.


LoadingAdd to solve later

Sponsored Links

More from my site

  • Is the Sum of a Nilpotent Matrix and an Invertible Matrix Invertible?Is the Sum of a Nilpotent Matrix and an Invertible Matrix Invertible? A square matrix $A$ is called nilpotent if some power of $A$ is the zero matrix. Namely, $A$ is nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix. Suppose that $A$ is a nilpotent matrix and let $B$ be an invertible matrix of […]
  • If Every Trace of a Power of a Matrix is Zero, then the Matrix is NilpotentIf Every Trace of a Power of a Matrix is Zero, then the Matrix is Nilpotent 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. Use the Jordan canonical form of the matrix $A$. We want […]
  • Nilpotent Matrix and Eigenvalues of the MatrixNilpotent Matrix and Eigenvalues of the Matrix An $n\times n$ matrix $A$ is called nilpotent if $A^k=O$, where $O$ is the $n\times n$ zero matrix. Prove the followings. (a) The matrix $A$ is nilpotent if and only if all the eigenvalues of $A$ is zero. (b) The matrix $A$ is nilpotent if and only if […]
  • Eigenvalues of Squared Matrix and Upper Triangular MatrixEigenvalues of Squared Matrix and Upper Triangular Matrix Suppose that $A$ and $P$ are $3 \times 3$ matrices and $P$ is invertible matrix. If \[P^{-1}AP=\begin{bmatrix} 1 & 2 & 3 \\ 0 &4 &5 \\ 0 & 0 & 6 \end{bmatrix},\] then find all the eigenvalues of the matrix $A^2$.   We give two proofs. The first version is a […]
  • Nilpotent Matrices and Non-Singularity of Such MatricesNilpotent Matrices and Non-Singularity of Such Matrices Let $A$ be an $n \times n$ nilpotent matrix, that is, $A^m=O$ for some positive integer $m$, where $O$ is the $n \times n$ zero matrix. Prove that $A$ is a singular matrix and also prove that $I-A, I+A$ are both nonsingular matrices, where $I$ is the $n\times n$ identity […]
  • Diagonalize the Upper Triangular Matrix and Find the Power of the MatrixDiagonalize the Upper Triangular Matrix and Find the Power of the Matrix Consider the $2\times 2$ complex matrix \[A=\begin{bmatrix} a & b-a\\ 0& b \end{bmatrix}.\] (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, determine the eigenvectors. (c) Diagonalize the matrix $A$. (d) Using the result of the […]
  • The Inverse Matrix of an Upper Triangular Matrix with VariablesThe Inverse Matrix of an Upper Triangular Matrix with Variables Let $A$ be the following $3\times 3$ upper triangular matrix. \[A=\begin{bmatrix} 1 & x & y \\ 0 &1 &z \\ 0 & 0 & 1 \end{bmatrix},\] where $x, y, z$ are some real numbers. Determine whether the matrix $A$ is invertible or not. If it is invertible, then find […]
  • True or False. Every Diagonalizable Matrix is InvertibleTrue or False. Every Diagonalizable Matrix is Invertible Is every diagonalizable matrix invertible?   Solution. The answer is No. Counterexample We give a counterexample. Consider the $2\times 2$ zero matrix. The zero matrix is a diagonal matrix, and thus it is diagonalizable. However, the zero matrix is not […]

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
Ohio State University exam problems and solutions in mathematics
Determine Whether the Following Matrix Invertible. If So Find Its Inverse Matrix.

Let A be the matrix \[\begin{bmatrix} 1 & -1 & 0 \\ 0 &1 &-1 \\ 0 & 0 &...

Close