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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Let $|A| = 6$, where $A$ is a $3 \times 3$ matrix. If
$|\text{adj}(3\text{adj}(A^2 \cdot \text{adj}(2A)))| = 2^m \cdot 3^n,; m,n \in \mathbb{N}$,
then $m + n$ is equal to ______.
Solution
$\text{adj}(2A) = 2^2 \text{adj}A$
$\Rightarrow \text{adj}(kA) = k^{n-1}\text{adj}(A)$
Now
$A^2(\text{adj}(2A)) = 4A(\text{adj}A)$
$= 4A|A|I_3 = 24A$
Now
$3\text{adj}(A^2(\text{adj}(2A))) = 3\text{adj}(24A)$
$= 3(24)^2 \text{adj}A$
Now
$|\text{adj}(3\text{adj}(A^2(\text{adj}(2A))))|$
$= |3(24)^2 \text{adj}A|^2$
$= |3 \cdot (24)^2|^2 \cdot |\text{adj}A|^2$
$= (3 \cdot 24^2)^2 \cdot (|A|^{2})^2$
$= 3^2 \cdot 24^4 \cdot 6^4$
$= 3^2 \cdot (2^3 \cdot 3)^4 \cdot (2 \cdot 3)^4$
$= 3^2 \cdot 2^{12} \cdot 3^4 \cdot 2^4 \cdot 3^4$
$= 2^{16} \cdot 3^{10}$
$\Rightarrow m + n = 16 + 46 = 62$
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