If the adjoint of a 3 times 3 matrix P is beg | vedclass

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(A, D) Let $P$ be a $3 	imes 3$ matrix. We are given that $\operatorname{adj}(P) = \begin{bmatrix} 1 & 4 & 4 \ 2 & 1 & 7 \ 1 & 1 & 3 \end{bmatrix}$

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$(A, B)$

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(A, D) Let $P$ be a $3 	imes 3$ matrix. We are given that $\operatorname{adj}(P) = \begin{bmatrix} 1 & 4 & 4 \ 2 & 1 & 7 \ 1 & 1 & 3 \end{bmatrix}$.We know the property $|\operatorname{adj}(P)| = |P|^{n-1}$,where $n$ is the order of the matrix.Here,$n = 3$,so $|\operatorname{adj}(P)| = |P|^{3-1} = |P|^2$.First,calculate the determinant of $\operatorname{adj}(P)$:$|\operatorname{adj}(P)| = 1(1 	imes 3 - 7 	imes 1) - 4(2 	imes 3 - 7 	imes 1) + 4(2 	imes 1 - 1 	imes 1)$$|\operatorname{adj}(P)| = 1(3 - 7) - 4(6 - 7) + 4(2 - 1)$$|\operatorname{adj}(P)| = 1(-4) - 4(-1) + 4(1)$$|\operatorname{adj}(P)| = -4 + 4 + 4 = 4$.Now,equate this to $|P|^2$:$|P|^2 = 4$$|P| = \pm 2$.Thus,the possible values of the determinant of $P$ are $2$ and $-2$,which correspond to options $(A)$ and $(D)$.

If the adjoint of a $3 \times 3$ matrix $P$ is $\begin{bmatrix} 1 & 4 & 4 \\ 2 & 1 & 7 \\ 1 & 1 & 3 \end{bmatrix}$,then the possible value$(s)$ of the determinant of $P$ is (are):

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