Let A be a 4 times 4 matrix and P be its adjo | vedclass

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(D) Given that $A$ is a $4 	imes 4$ matrix,so $n = 4$. The adjoint matrix $P = \operatorname{adj}(A)$. We know that $|\operatorname{adj}(A)| = |A|^{n

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$\pm 4$

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(D) Given that $A$ is a $4 	imes 4$ matrix,so $n = 4$. The adjoint matrix $P = \operatorname{adj}(A)$. We know that $|\operatorname{adj}(A)| = |A|^{n-1}$,so $|P| = |A|^{4-1} = |A|^3$. Given $|P| = |\frac{A}{2}|$. Since $A$ is a $4 	imes 4$ matrix,$|\frac{A}{2}| = \frac{1}{2^4} |A| = \frac{1}{16} |A|$. Equating the two expressions: $|A|^3 = \frac{1}{16} |A|$. This implies $|A|^3 - \frac{1}{16} |A| = 0$,so $|A|(|A|^2 - \frac{1}{16}) = 0$. Since $|A| 
eq 0$ (as $A^{-1}$ exists),$|A|^2 = \frac{1}{16}$,which gives $|A| = \pm \frac{1}{4}$. Finally,$|A^{-1}| = \frac{1}{|A|} = \frac{1}{\pm 1/4} = \pm 4$.

Let $A$ be a $4 \times 4$ matrix and $P$ be its adjoint matrix. If $|P|=\left|\frac{A}{2}\right|$,then $\left|A^{-1}\right|=$

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