Question:
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 corret entry in List-II.List - I List - II (P) The number of matrices M = ( aij )3×3 with all entries in T such that Ri = Cj = 0 for all i, j, is (1) 1 (Q) The number of symmetric matrices M = ( aij )3×3 with all entries in T such that Cj = 0 for all j, is (2) 12 (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
\(\left\{\begin{pmatrix} x \\ y \\ z \end{pmatrix}:x,y,z\in \R, M\begin{pmatrix} x \\ y \\ z \end{pmatrix}=\begin{pmatrix} a_{12} \\ 0 \\ -a_{23} \end{pmatrix}\right\}\) is (3) infinite (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 is (4) 6 (5) 0
The correct option is
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 corret entry in List-II.
The correct option is
Match each entry in List-I to the corret entry in List-II.
| List - I | List - II | ||
| (P) | The number of matrices M = ( aij )3×3 with all entries in T such that Ri = Cj = 0 for all i, j, is | (1) | 1 |
| (Q) | The number of symmetric matrices M = ( aij )3×3 with all entries in T such that Cj = 0 for all j, is | (2) | 12 |
| (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 \(\left\{\begin{pmatrix} x \\ y \\ z \end{pmatrix}:x,y,z\in \R, M\begin{pmatrix} x \\ y \\ z \end{pmatrix}=\begin{pmatrix} a_{12} \\ 0 \\ -a_{23} \end{pmatrix}\right\}\) is | (3) | infinite |
| (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 is | (4) | 6 |
| (5) | 0 | ||
Updated On: Mar 7, 2025
- (P) → (4) (Q) → (2) (R) → (5) (S) → (1)
- (P) → (2) (Q) → (4) (R) → (1) (S) → (5)
- (P) → (2) (Q) → (4) (R) → (3) (S) → (5)
- (P) → (1) (Q) → (5) (R) → (3) (S) → (4)
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The Correct Option is C
Solution and Explanation
1. For (P):
- The matrix M = (aij)3×3 has entries in T = {α, β} such that row sums (Ri) and column sums (Cj) are zero for all i, j.
- Since each row and column sum to zero, there are exactly 12 matrices that satisfy these constraints.
(P) → (2).
2. For (Q):
- The matrix M is symmetric, meaning aij = aji, and has entries in T = {α, β}.
- Additionally, column sums (Cj) are zero for all j.
- The number of symmetric matrices satisfying these conditions is exactly 6.
(Q) → (4).
3. For (R):
- The matrix M = (aij)3×3 is skew-symmetric, meaning aij = -aji for all i, j.
- The entries aij belong to T for i > j.
- Since the diagonal elements in a skew-symmetric matrix are always zero and off-diagonal elements can be freely chosen from T, there are infinitely many possible matrices.
(R) → (3).
4. For (S):
- The matrix M = (aij)3×3 has entries in T = {α, β} and satisfies the row sum constraint (Ri = 0) for all i.
- Since each row sum is zero, the rows are linearly dependent, meaning the determinant of M must be zero.
- Absolute value of determinant: 0.
(S) → (5).
Final Answer:
(P) → (2), (Q) → (4), (R) → (3), (S) → (5)
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