QJEEM24Ma7096: JEE Main 2024 Mathematics - Class 12 JEEM24Ma7096 Solution: Let be a real matrix and be the identity matrix of order 2. If…

$|A-x I|=0$

Roots are $-$ 1 and 3

Sum of roots $=\operatorname{tr}(A)=2$

Product of roots $=|\mathrm{A}|=-3$

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

We have $\mathrm{a}+\mathrm{d}=2$

$\mathrm{ad}-\mathrm{bc}=-3$

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

We need $a^2+b c+b c+d^2$

$\begin{aligned}& =a^2+2 b c+d^2 \\& =(a+d)^2-2 a d+2 b c \\& =4-2(a d-b c) \\& =4-2(-3) \\& =4+6 \\& =10\end{aligned}$