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

Problem 582

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 the same size as $A$.
Is the matrix $B-A$ invertible? If so prove it. Otherwise, give a counterexample.

Solution.

We claim that the matrix $B-A$ is not necessarily invertible.
Consider the matrix
$A=\begin{bmatrix} 0 & -1 \\0& 0 \end{bmatrix}.$ This matrix is nilpotent as we have
$A^2=\begin{bmatrix} 0 & -1 \\0& 0 \end{bmatrix} \begin{bmatrix} 0 & -1 \\0& 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\0& 0 \end{bmatrix}.$

Also consider the matrix
$B=\begin{bmatrix} 1 & 0 \\1& 1 \end{bmatrix}.$ Since the determinant of the matrix $B$ is $1$, it is invertible.

So the matrix $A$ and $B$ satisfy the assumption of the problem.
However the matrix
$B-A=\begin{bmatrix} 1 & 1 \\1& 1 \end{bmatrix}$ is not invertible as its determinant is $0$.
Hence we found a counterexample.

Related Question.

Here is another problem about a nilpotent matrix.

Problem.
Let $A$ be an $n\times n$ nilpotent matrix. Then prove that $I-A, I+A$ are both nonsingular matrices, where $I$ is the $n\times n$ identity matrix.

The solution is given in the post ↴
Nilpotent Matrices and Non-Singularity of Such Matrices

More from my site

• Is the Product of a Nilpotent Matrix and an Invertible Matrix Nilpotent? 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 $P$ […]
• Every Diagonalizable Nilpotent Matrix is the Zero Matrix Prove that if $A$ is a diagonalizable nilpotent matrix, then $A$ is the zero matrix $O$.   Definition (Nilpotent Matrix) A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$. Proof. Main Part Since $A$ is […]
• Nilpotent 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 […]
• True 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 […]
• 12 Examples of Subsets that Are Not Subspaces of Vector Spaces Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace. (1) $S_1=\left \{\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}$ in […]
• Normal Nilpotent Matrix is Zero Matrix A complex square ($n\times n$) matrix $A$ is called normal if $A^* A=A A^*,$ where $A^*$ denotes the conjugate transpose of $A$, that is $A^*=\bar{A}^{\trans}$. A matrix $A$ is said to be nilpotent if there exists a positive integer $k$ such that $A^k$ is the zero […]
• Find Inverse Matrices Using Adjoint Matrices Let $A$ be an $n\times n$ matrix. The $(i, j)$ cofactor $C_{ij}$ of $A$ is defined to be $C_{ij}=(-1)^{ij}\det(M_{ij}),$ where $M_{ij}$ is the $(i,j)$ minor matrix obtained from $A$ removing the $i$-th row and $j$-th column. Then consider the $n\times n$ matrix […]
• For Which Choices of $x$ is the Given Matrix Invertible? Determine the values of $x$ so that the matrix $A=\begin{bmatrix} 1 & 1 & x \\ 1 &x &x \\ x & x & x \end{bmatrix}$ is invertible. For those values of $x$, find the inverse matrix $A^{-1}$.   Solution. We use the fact that a matrix is invertible […]

1 Response

1. 11/18/2017

[…] the post ↴ Is the Sum of a Nilpotent Matrix and an Invertible Matrix Invertible? for the […]

The Subspace of Linear Combinations whose Sums of Coefficients are zero

Let $V$ be a vector space over a scalar field $K$. Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$...

Close