Proving or disproving matrix $A+B$ is invertible

Question

Given $A, B \in M_n (\Bbb F)$, where $A$ is $k$-nilpotent and $B$ is invertible, is $A+B$ also invertible?

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I was having trouble on how to prove this, and then I thought maybe this statement is incorrect, but couldn't find a counter example. Perhaps someone can assist?

Answer 1

This is not true in general. Take $$ A = \begin{pmatrix} 0 &1 \\ 0 & 0 \end{pmatrix}, \quad B= \begin{pmatrix} 0 &-1 \\ 1& 0 \end{pmatrix}. $$ However, $I+A$ is always invertible for nilpotent $A$. The same holds for $A+B$ with invertible $B$ if $A$ and $B$ commute: If $A$ and $B$ commute, then $A$ and $B^{-1}$ commute, which implies that $B^{-1}A$ is nilpotent, moreover $I-(-B^{-1}A)$ is invertible with $$ (I-(-B^{-1}A))^{-1} = \sum_{i=0}^{k-1} (-B^{-1}A)^{i} $$ which implies $$ (A+B)^{-1} = B^{-1}(I+B^{-1}A)^{-1}. $$

Answer 2

Hint: Consider the case $k = 2$ with $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$. Can you find a matrix $B$ with linearly independent columns such that the columns of $A + B$ become linearly dependent?

Answer 3

A counterexample for $n=2$ by taking

$$A=\begin{pmatrix}0&1\\0&0\end{pmatrix}$$ and $$B=\begin{pmatrix}1&0\\1&1\end{pmatrix}$$