# Find two different 2 times 2 matrices A such that A^2=O .

Nannie Mack 2020-10-27 Answered
Find two different $2×2$ matrices A such that ${A}^{2}=O$ .
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Step 1
Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ be the $2×2$ matrix.
Step 2
Then,
${A}^{2}=\left[\begin{array}{cc}{a}^{2}+bc& \left(a+d\right)b\\ \left(a+d\right)c& {d}^{2}+bc\end{array}\right]$
Given $A\ne 0$
So, find the values of a, b, c, d so that each element of ${A}^{2}=0$
That is, when
a^2+bc=0
(a+d)b=0
(a+d)c=0
d^2+bc=0
so,
$\left(a+d\right)b=0⇒a+d=0⇒a=-d$
$ad-bc=0⇒ad=bc$
Step 3
For instance, put a=-d=1, b=-c=1 or put a=d=0, b=0, c=1. Then the matrices will be $\left[\begin{array}{cc}1& 1\\ -1& -1\end{array}\right],\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$
Jeffrey Jordon