Find a example of $A$ be $4 \times 4$ matrix such that $A$ has rank $2$ but $A^2 =0 $?

Question

Find a example of $A$ be $4 \times 4$ matrix such that $A$ has rank $2$ but $A^2 =0 $?

My attempt :

$$A=\begin{bmatrix} 0 & 0 & 1 &0\\0 & 0 & 1 & 0\\0 &0 &0 &0 \\0 &0 &0 &0 \\\end{bmatrix}$$

Is it correct ??

Any hints/solution will be apprecaited

thanks u

Answer 1

Your first 2 lines are not independent thus the rank is 1 and not 2. Look at $$A=\begin{bmatrix} 0 & 0 & 1 &0\\0 & 0 & 0 & 0\\0 &0 &0 &0 \\0 &1 &0 &0 \\\end{bmatrix}$$