Question

Suppose $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is a real matrix with nonzero entries, $ad–bc=0$, and $A^2=A$. Then $a+d$ equals

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is A 1

Given :
$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and $ad–bc=0$, and $A^2=A$

$\Rightarrow A^2=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}{a}^2+bc& ab+bd\\ ac+cd& bc+d^2\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
$\therefore a^2+bc=a\Rightarrow a(a+d)=a$
[$\because ad-bc=0$]
Hence $a+d=1$