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.

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.

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