Matrices & Determinants Question - None

Question

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 $[\begin{matrix} 1 & a & b \\ ω & 1 & c \\ ω^{2} & ω & 1 \end{matrix}]$ , 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 $M [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] = [\begin{matrix} - 1 \\ 2 \\ 3 \end{matrix}], M [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix}]$ and $M [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 12 \end{matrix}]$ . 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 $A [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]$ has exactly two distinct solutions, is

(D) Let k be a positive real number and let $A = [\begin{matrix} 2 k - 1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & - 2 k \\ - 2 \sqrt{k} & 2 k & - 1 \end{matrix}]$ and $B = [\begin{matrix} 0 & 2 k - 1 & \sqrt{k} \\ 1 - 2 k & 0 & 2 \sqrt{k} \\ - \sqrt{k} & - 2 \sqrt{k} & 0 \end{matrix}]$ . 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

Accepted Answer

(A) The determinant of the matrix is 1(1-cω) - a(ω-cω²) + b(ω²-ω²) = 1 - cω - aω + acω². Since a, b, c ∈ {ω, ω²}, we test combinations to find non-singular matrices. The number of such matrices is 2. (B) By solving the system of equations for the columns of M, we find the diagonal entries and their sum is 9. (C) A system of linear equations Ax=b can have 0, 1, or infinitely many solutions. It cannot have exactly two solutions. Thus, the number of such matrices is 0. (D) Using det(adj A) = (det A)² and solving the equation leads to [k] = 4. Matching these gives A→S, B→R, C→P, D→Q.

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