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

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(A) Let $M$ be a $3 \times 3$ matrix with real entries such that $M^2 = X$. Then $\det(M^2) = \det(X)$, which implies $(\det(M))^2 = \det(X)$. Since $M$ has real entries, $\det(M)$ is a real number, so $(\det(M))^2 \ge 0$. Thus, for a matrix $X$ to be a square of a real matrix, its determinant must be non-negative.For $A$: $\det(A) = (1)(1)(-1) = -1 For $B$: $\det(B) = (-1)(-1)(-1) = -1 For $C$: $\det(C) = (1)(1)(1) = 1 > 0$. It is the square of the identity matrix $I^2 = I$.For $D$: $\det(D) = (1)(-1)(-1) = 1 > 0$. 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 $3 \times 3$ matrices.

Class 12 Mathematics 3-4. Matrices and Determinants Mix Examples-Determinants and Matrices

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Which of the following matrices are $NOT$ the square of a $3 \times 3$ matrix with real entries? Let the matrices be defined as:
$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$, $B = \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$, $C = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$, $D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$.

IIT 2017 All IIT

A
$A, B$
B
$A, C$
C
$A, D$
D
$A, B, C$

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3-4. Matrices and Determinants

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Mix Examples-Determinants and Matrices

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IIT 2017 Paper All IIT years →

If $A = \begin{bmatrix} 0 & -\tan \frac{\alpha}{2} \\ \tan \frac{\alpha}{2} & 0 \end{bmatrix}$ and $I$ is the identity matrix of order $2$,show that $I+A = (I-A) \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$.

Difficult

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Let $\alpha \in(0, \infty)$ and $A=\begin{bmatrix} 1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2 \end{bmatrix}$. If $\operatorname{det}(\operatorname{adj}(2A-A^{T}) \cdot \operatorname{adj}(A-2A^{T}))=2^8$,then $(\operatorname{det}(A))^2$ is equal to:

DifficultJEE MAIN 2024

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If the matrix $A = \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix}$ is expressed as the sum of a symmetric matrix $B$ and a skew-symmetric matrix $C$, then which of the following relations is correct?

DifficultMHT CET 2026

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Let $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ and $P = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$. Let $Q = \begin{bmatrix} x & y \\ z & 4 \end{bmatrix}$ for some non-zero real numbers $x, y$,and $z$,for which there exists a $2 \times 2$ matrix $R$ with all entries being non-zero real numbers,such that $QR = RP$. Then which of the following statements is (are) true?
$(A)$ $|Q-2I| = 0$
$(B)$ $|Q-6I| = 12$
$(C)$ $|Q-3I| = 15$
$(D)$ $yz = 2$

IIT 2025

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Let $A = \begin{bmatrix} 2 & b & 1 \\ b & b^2+1 & b \\ 1 & b & 2 \end{bmatrix}$ where $b > 0$. Then the minimum value of $\frac{\det(A)}{b}$ is

DifficultJEE MAIN 2019

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If $A = \begin{bmatrix} 0 & -\tan \frac{\alpha}{2} \\ \tan \frac{\alpha}{2} & 0 \end{bmatrix}$ and $I$ is the identity matrix of order $2$,show that $I+A = (I-A) \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$.

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