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For a non-singular $n \times n$ matrix $A$, the relation between $\det(\operatorname{adj} A)$ and $\det A$ is:
A$|\operatorname{adj} A| = |A|$
B$|\operatorname{adj} A| = |A|^{n}$
C$|\operatorname{adj} A| = |A|^{n-1}$
D$|\operatorname{adj} A| = \dfrac{1}{|A|}$
Answer & Solution
Correct answer: C. $|\operatorname{adj} A| = |A|^{n-1}$
From $A\,\operatorname{adj}(A) = |A| I_n$, taking determinants: $|A| \cdot |\operatorname{adj}(A)| = |A|^n$. Hence $|\operatorname{adj}(A)| = |A|^{n-1}$ when $|A| \ne 0$.
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