Which of the following is(are) NOT the square | vedclass

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(A) Let $M$ be a $3 	imes 3$ matrix with real entries such that $M^2 = A$. Then $\det(M^2) = \det(A)$,which implies $(\det(M))^2 = \det(A)$. Since $M

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$A, B$

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(A) Let $M$ be a $3 	imes 3$ matrix with real entries such that $M^2 = A$. Then $\det(M^2) = \det(A)$,which implies $(\det(M))^2 = \det(A)$. Since $M$ has real entries,$\det(M)$ is a real number,so $(\det(M))^2 \ge 0$. Thus,for $A$ to be a square of a real matrix,its determinant must be non-negative.For option $A$: $\det(A) = \det \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & -1 \end{bmatrix} = -1 For option $B$: $\det(B) = \det \begin{bmatrix} -1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & -1 \end{bmatrix} = -1 For option $C$: $\det(C) = 1 > 0$. It is the square of the identity matrix $I^2 = I$.For option $D$: $\det(D) = \det \begin{bmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & -1 \end{bmatrix} = 1 > 0$. It is the square of the matrix $\begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & -1 & 0 \end{bmatrix}$.Therefore,$A$ and $B$ are not squares of real $3 	imes 3$ matrices.

Which of the following is(are) $NOT$ the square of a $3 \times 3$ matrix with real entries?

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