MCQ
Suppose $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ is a real matrix with non-zero entries, $\alpha d-b c=0$ and $A^2=A$. Then, $a + d$ equals
- ✓$1$
- B$2$
- C$3$
- D$4$
We have
$\begin{aligned} A &=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \\ A^2 &=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \\ A^2 &=\left[\begin{array}{ll}a^2+b c & a b+b d \\ a c+c d & b c+d^2\end{array}\right] \end{aligned}$
Given, $A^2=A$ and $a d-b c=0$
$\therefore\left[\begin{array}{ll} a^2+b c & a b+b d \\ a c+c d & b c+d^2 \end{array}\right]=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$
$a b+b d =b$
$b(a+d) =b$
$a+d =1$
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$x^{2}-\left(5+3 \sqrt{\log _{3} 5}-5 \sqrt{\log _{5} 3}\right)x+3\left(3^{\left(\log _{3} 5\right)^{\frac{1}{3}}}-5^{\left(\log _{5} 3\right)^{\frac{2}{3}}}-1\right)=0$
then the equation, whose roots are $\alpha+\frac{1}{\beta} \text { and } \beta+\frac{1}{\alpha} \text {, }$