How to get matrix $A$ knowing that $A^2 = O_2$?

Question

Given a matrix $$A = \left[\begin{matrix} a & b \\ c & d \\ \end{matrix}\right],$$ how do you solve the equations of $$A^2 = \left[\begin{matrix} a^2 + bc & b(a+d) \\ c(a+d) & d^2 + cb \\ \end{matrix}\right] = O_2?$$

Answer 1

Your system is$$\left\{\begin{array}{l}a^2+bc=0\\b(a+d)=0\\c(a+d)=0\\d^2+bc=0.\end{array}\right.$$If $a=-d$, this system becomes simply $a^2+bc=0$.

If $d\neq-a$, then $b=c=0$. And then it follows from the first and fourth equation that $a=d=0$ too.

So, the solutions of your matrix equation are the matrices of the form $\left[\begin{smallmatrix}a&b\\c&-a\end{smallmatrix}\right]$ such that $a^2+bc=0$.

Answer 2

$A\in M_2(K)$ is nilpotent if and only if $\det(A)=\operatorname{tr}(A)=0$ by Cayley-Hamilton. This means $a+d=0$ and $a^2+bc=0$.