MathDB

Let

M,N\in M_2(\mathbb C)

be two nonzero matrices such that

M^2=N^2=0_2\text{ and }MN+NM=I_2

where

0_2

is the

2\times2

zero matrix and

I_2

the

2\times2

unit matrix. Prove that there is an invertible matrix

A\in M_2(\mathbb C)

such that

M=A\begin{pmatrix}0&1\\0&0\end{pmatrix}A^{-1}\text{ and }N=A\begin{pmatrix}0&0\\1&0\end{pmatrix}A^{-1}.

Problem Statement