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3 and 4 .Determinants and Matrices
hard
ધારોકે $A =\left[ a _{ ij }\right]_{2 \times 2}$, જ્યાં પ્રત્યેક $i , j$ માટ $a _{ ij } \neq 0$ અને $A ^2= I$.ધારોકે $A$ ના તમામ વિકર્ણી ઘટકોનો સરવાળો $a$ છે અને $b =| A |$. તો $3 a ^2+4 b ^2=.......$
A
$7$
B
$14$
C
$3$
D
$4$
(JEE MAIN-2023)
Solution
$\begin{aligned}& \text { Let } A=\left[\begin{array}{ll} p & l \\r & s\end{array}\right] \\& A^2=\left[\begin{array}{ll} p ^2+ qr & pq + qs \\pr + rs & qs + s ^2\end{array}\right] \\& \Rightarrow p ^2+ qr =1(1) pq + qs =0 \Rightarrow q ( p + s )=0 \\& \Rightarrow s ^2+ qr =1(2) pr + rs =0 \Rightarrow r(p+s)=0\end{aligned}$
Equation $(1)$ – equation $(2)$
$p ^2= s ^2 \Rightarrow p + s =0$
Now $3 a^2+4 b^2$
$=3(p+s)^2+4(p s-q r)^2$
$=3.0+4\left(-p^2-q r\right)^2=4\left(p^2+q r\right)^2=4$
Standard 12
Mathematics
