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QUESTION Show that every $2\times2$ matrix $A$ for which $A^2=0$
has det$A$=0 and tr$A$=0.


ANSWER

 $\left[\begin{array}{cc}a&b\\c&d\end{array}\right]
\left[\begin{array}{cc}a&b\\c&d\end{array}\right]=
\left[\begin{array}{cc}a^2+bc&ab+bd\\ca+dc&cb+d^2\end{array}\right]=
\left[\begin{array}{cc}a^2+bc&b(a+d)\\c(a+d)&bc+d^2\end{array}\right]$

If $b\neq 0$ and $c\neq 0$ then $a+d=$tr$A=0$ and $a^2+bc=0$.
Hence det$A=ad-bc=-a^2-bc=0$;or use det$(A^2)=($det$A)^2$. [The
cases where either $b$ or $c$=0 need to be considered separately.]

The general matrix with $A^2=0$ (and $b\neq 0$) can be written

$\left[\begin{array}{cc}a&b\\-\frac{a^2}{b}&-a\end{array}\right]$


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