Question

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, if $ad-bc=0$, $A^2=A$.Find $a+d$.

Answer 1

$A^2=A$

$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$\left[\begin{array}{cc}{a}^2+bc& ab+bd\\ ac+cd& bc+d^2\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$ac+cd=c$

$c(a+d)=c$

$a+d=1$.