Column-I(A) Let ω ≠ 1 be a cube root of unity and S be the set of all non-singular matrices of the form 1abω1cω2ω1, where each of a, b and c is either ω or ω2. Then the number of distinct matrices in the set S is-(B) Let M be 3 × 3 matrix satisfying M100=-123,M1-10=11-1 and M111=0012. Then the sum of the diagonal entries of M is(C) The number of 3 × 3 matrices A whose entries are either 0 or 1 and for which the system Axyz=100 has exactly two distinct solutions, is(D) Let k be a positive real number and let A=2k-12k2k2k1-2k-2k2k-1 and B=02k-1k1-2k02k-k-2k0. If det(adj A) + det(adj B) = 106, then [k] is equal to [Note: adj M denotes the adjoint of a square matrix M and [k] denotes the largest integer less than or equal to k].Column-II(P) 0(Q) 4(R) 9(S) 2
Let $R = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ be a non-zero $3 \times 3$ matrix, where $x \sin \theta = y \sin \left( \theta + \frac{2\pi}{3} \right) = z \sin \left( \theta + \frac{4\pi}{3} \right) \neq 0, \theta \in (0, 2\pi)$. For a square matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of $M$. Then, among the statements: (I) Trace $(R) = 0$ (II) If trace $(\text{adj}(\text{adj}(R))) = 0$, then $R$ has exactly one non-zero entry.
Let $M=(a_{ij}), i,j \in \{1, 2, 3\}$, be the $3 \times 3$ matrix such that $a_{ij} = 1$ if $j+1$ is divisible by $i$, otherwise $a_{ij} = 0$. Then which of the following statements is (are) true ?(A) $M$ is invertible(B) There exists a nonzero column matrix $\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$ such that $M \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} = \begin{pmatrix} -a_1 \\ -a_2 \\ -a_3 \end{pmatrix}$(C) The set $\{X \in \mathbb{R}^3 : MX=0\} \neq \{0\}$, where $0 = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$(D) The matrix $(M - 2I)$ is invertible, where $I$ is the $3 \times 3$ identity matrix
Let α and β be the distinct roots of the equation x2 + x - 1 = 0. Consider the set T = {1, α, β}. For a 3 × 3 matrix M = (aij)3×3, define Ri = ai1 + ai2 + ai3 and Cj = a1j + a2j + a3j for i = 1, 2, 3 and j = 1, 2, 3. Match each entry in List-I to the correct entry in List-II.List-I(P) The number of matrices M = (aij)3×3 with all entries in T such that Ri = Cj = 0 for all i, j, is(Q) The number of symmetric matrices M = (aij)3×3 with all entries in T such that Cj = 0 for all j, is(R) Let M = (aij)3×3 be a skew symmetric matrix such that aij ∈ T for i > j. Then the number of elements in the set { (x, y, z) ∈ R3 : M(x, y, z)T = 0 } is(S) Let M = (aij)3×3 be a matrix with all entries in T such that Ri = 0 for all i. Then the absolute value of the determinant of M isList-II(1) 1(2) 12(3) infinite(4) 6(5) 0
Consider the following statementsStatement-1 : Given $A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 4 & 1 \\ 2 & 3 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix}$. If $BPA = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$, then $\lambda$ denotes sum of elements of $P$.Statement-2 : Let $\mu$ denote the sum of elements of the matrix $A$ satisfying the matrix equation, $\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} A \begin{bmatrix} 3 & 2 \\ 5 & -3 \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 3 & -1 \end{bmatrix}$Statement-3 : Given that $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 2 & 1 \\ 1 & -1 & 3 \end{bmatrix}$, $C = \begin{bmatrix} 2 & 1 & 1 \\ 2 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$, $D = \begin{bmatrix} 10 \\ 13 \\ 9 \end{bmatrix}$ and that $Cb = D$. If $AX = b$, then $v$ denotes the sum of elements of $X$.Then, which of the following options is/are correct?(A) $2\lambda - 19\mu = 3$(B) $38\mu + 15v = 3$(C) $10v + 8\mu = 2$(D) $\lambda + 19\mu + 38\mu = 0$