(D) Let $P = A^2 B^2 - B^2 A^2$. Taking the transpose of $P$: $P^T = (A^2 B^2 - B^2 A^2)^T = (A^2 B^2)^T - (B^2 A^2)^T$. Using the property $(XY)^T =

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(D) Let $P = A^2 B^2 - B^2 A^2$. Taking the transpose of $P$: $P^T = (A^2 B^2 - B^2 A^2)^T = (A^2 B^2)^T - (B^2 A^2)^T$. Using the property $(XY)^T = Y^T X^T$: $P^T = (B^2)^T (A^2)^T - (A^2)^T (B^2)^T$. Since $A$ is symmetric $(A^T = A)$ and $B$ is skew-symmetric $(B^T = -B)$,we have $(A^2)^T = (A^T)^2 = A^2$ and $(B^2)^T = (B^T)^2 = (-B)^2 = B^2$. Substituting these: $P^T = B^2 A^2 - A^2 B^2 = -(A^2 B^2 - B^2 A^2) = -P$. Since $P^T = -P$,$P$ is a skew-symmetric matrix. For any skew-symmetric matrix $P$ of odd order $n$ (here $n=3$),the determinant $\det(P) = 0$. Since $\det(P) = 0$,the system $PX = 0$ has a non-trivial solution,implying it has infinitely many solutions.

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

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Let $A$ and $B$ be $3 \times 3$ real matrices such that $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix. Then the system of linear equations $(A^2 B^2 - B^2 A^2) X = 0$,where $X$ is a $3 \times 1$ column matrix of unknown variables and $0$ is a $3 \times 1$ null matrix,has:

MHT CET 2024 All MHT CET

A
a unique solution
B
exactly two solutions
C
no solution
D
infinitely many solutions

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Mix Examples-Determinants and Matrices

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If $\alpha , \beta \neq 0$ and $f(n) = \alpha^n + \beta^n$ and $\begin{vmatrix} 3 & 1 + f(1) & 1 + f(2) \\ 1 + f(1) & 1 + f(2) & 1 + f(3) \\ 1 + f(2) & 1 + f(3) & 1 + f(4) \end{vmatrix} = K(1 - \alpha)^2 (1 - \beta)^2 (\alpha - \beta)^2$,then $K = \dots$

DifficultJEE MAIN 2014

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If $a_{r} = \cos \frac{2 r \pi}{9} + i \sin \frac{2 r \pi}{9}$,$r = 1, 2, 3, \ldots$,$i = \sqrt{-1}$,then the determinant $\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|$ is equal to:

MediumJEE MAIN 2021

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Consider the following linear equations:
$ax+by+cz=0$,$bx+cy+az=0$,$cx+ay+bz=0$
Match the conditions/expressions in Column $I$ with statements in Column $II$:
Column $I$ Column $II$
$(A)$ $a+b+c \neq 0$ and $a^2+b^2+c^2=ab+bc+ca$ $(p)$ The equations represent planes meeting only at a single point.
$(B)$ $a+b+c=0$ and $a^2+b^2+c^2 \neq ab+bc+ca$ $(q)$ The equations represent the line $x=y=z$.
$(C)$ $a+b+c \neq 0$ and $a^2+b^2+c^2 \neq ab+bc+ca$ $(r)$ The equations represent identical planes.
$(D)$ $a+b+c=0$ and $a^2+b^2+c^2=ab+bc+ca$ $(s)$ The equations represent the whole of the three-dimensional space.

IIT 2007

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Let $A = [a_{ij}]$,where $a_{ij} \in \mathbb{Z} \cap [0, 4]$ and $1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2, 13)$ is $........$.

DifficultJEE MAIN 2023

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If $A = \begin{bmatrix} \cos \theta & i \sin \theta \\ i \sin \theta & \cos \theta \end{bmatrix}$,$\theta = \frac{\pi}{24}$ and $A^{5} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$,where $i = \sqrt{-1}$,then which one of the following is not true?

DifficultJEE MAIN 2020

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Column $I$	Column $II$
$(A)$ $a+b+c \neq 0$ and $a^2+b^2+c^2=ab+bc+ca$	$(p)$ The equations represent planes meeting only at a single point.
$(B)$ $a+b+c=0$ and $a^2+b^2+c^2 \neq ab+bc+ca$	$(q)$ The equations represent the line $x=y=z$.
$(C)$ $a+b+c \neq 0$ and $a^2+b^2+c^2 \neq ab+bc+ca$	$(r)$ The equations represent identical planes.
$(D)$ $a+b+c=0$ and $a^2+b^2+c^2=ab+bc+ca$	$(s)$ The equations represent the whole of the three-dimensional space.

Let $A$ and $B$ be $3 \times 3$ real matrices such that $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix. Then the system of linear equations $(A^2 B^2 - B^2 A^2) X = 0$,where $X$ is a $3 \times 1$ column matrix of unknown variables and $0$ is a $3 \times 1$ null matrix,has:

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If $\alpha , \beta \neq 0$ and $f(n) = \alpha^n + \beta^n$ and $\begin{vmatrix} 3 & 1 + f(1) & 1 + f(2) \\ 1 + f(1) & 1 + f(2) & 1 + f(3) \\ 1 + f(2) & 1 + f(3) & 1 + f(4) \end{vmatrix} = K(1 - \alpha)^2 (1 - \beta)^2 (\alpha - \beta)^2$,then $K = \dots$

If $a_{r} = \cos \frac{2 r \pi}{9} + i \sin \frac{2 r \pi}{9}$,$r = 1, 2, 3, \ldots$,$i = \sqrt{-1}$,then the determinant $\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|$ is equal to:

Let $A = [a_{ij}]$,where $a_{ij} \in \mathbb{Z} \cap [0, 4]$ and $1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2, 13)$ is $........$.

If $A = \begin{bmatrix} \cos \theta & i \sin \theta \\ i \sin \theta & \cos \theta \end{bmatrix}$,$\theta = \frac{\pi}{24}$ and $A^{5} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$,where $i = \sqrt{-1}$,then which one of the following is not true?

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