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3 and 4 .Determinants and Matrices
hard
ધારોકે $\alpha$ અને $\beta$ વાસ્તવિક સંખ્યાઓ છે. $3 \times 3$ શ્રેણિક $A$ એવો છે કે જેથી $A^2=3 A+\alpha I$. જો $A^4=21 A+\beta I$ હોય, તો $..........$
A
$\alpha=1$
B
$\alpha=4$
C
$\beta=8$
D
$\beta=-8$
(JEE MAIN-2023)
Solution
$A ^2=3 A +\alpha I$
$A ^3=3 A ^2+\alpha A$
$A ^3=3(3 A +\alpha I )+\alpha A$
$A ^3=9 A +\alpha A +3 \alpha I$
$A ^4=(9+\alpha) A ^2+3 \alpha A$
$=(9+\alpha)(3 A +\alpha I )+3 \alpha A$
$= A (27+6 \alpha)+\alpha(9+\alpha)$
$\Rightarrow 27+6 \alpha=21 \Rightarrow \alpha=-1$
$\Rightarrow \beta=\alpha(9+\alpha)=-8$
Standard 12
Mathematics
