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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
If $A$ is a square matrix of order $3$ such that det(A) = 3 and
$\det(\text{adj}(-4,\text{adj}(-3 \ \text{adj}(3 \ \text{adj}((2A)^{-1}))))) = 2^m 3^n$,
then m + 2n is equal to:
Solution
Key properties: $\det(\text{adj}(X)) = (\det X)^2$
$\det(kX) = k^3 \det(X)$
$\det(X^{-1}) = \frac{1}{\det(X)}$
Step 1:
$\det((2A)^{-1}) = \frac{1}{2^3 \cdot 3} = 2^{-3} \cdot 3^{-1}$
Now track powers only
Each $\text{adj}$ ⇒ square powers
Each scalar $k$ ⇒ multiply by $k^3$
Each scalar $k$ ⇒ multiply by $k^3$
Start: $2^{-3} 3^{-1}$
Apply $\text{adj}$ ⇒ square
$2^{-6} 3^{-2}$
Multiply by $3$ ⇒ $3^3$
$2^{-6} 3^{1}$
Apply $\text{adj}$ ⇒ square
$2^{-12} 3^{2}$
Multiply by $-3$ ⇒ $(-3)^3 = -27$ (ignore sign for powers)
$2^{-12} 3^{5}$
Apply $\text{adj}$ ⇒ square
$2^{-24} 3^{10}$
Multiply by $-4$ ⇒ $(-4)^3 = -64 = 2^6$
$2^{-18} 3^{10}$
Apply $\text{adj}$ ⇒ square
$2^{-36} 3^{20}$
So, $m=-36,; n=20$
$m + 2n = -36 + 40 = 4$
Final Answer: $ \boxed{4} $
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