Question

Let $\alpha$ and $\beta$ be the distinct roots of the equation $\mathrm{x}^{2}+\mathrm{x}-1=0$. Consider the set $\mathrm{T}=\{1, \alpha, \beta\}$. For a $3 \times 3$ matrix $\mathrm{M}=\left(\mathrm{a}_{\mathrm{ij}}\right)_{3 \times 3}$, define $\mathrm{R}_{\mathrm{i}}=\mathrm{a}_{\mathrm{i} 1}+\mathrm{a}_{\mathrm{i} 2}+\mathrm{a}_{\mathrm{i} 3}$ and $\mathrm{C}_{\mathrm{j}}=\mathrm{a}_{1 \mathrm{i}}+\mathrm{a}_{2 \mathrm{j}}+\mathrm{a}_{3 \mathrm{j}}$ for $\mathrm{i}=1,2,3$ and $\mathrm{j}=1,2,3$.
Match each entry in List-I to the correct entry in List-II.

| List-I | | List-II | |
| --- | --- | --- | --- |
| (P) | The number of matrices $\mathrm{M}=\left(\mathrm{a}_{\mathrm{ij}}\right)_{3 \times 3}$ with all entries in T such that $\mathrm{R}_{\mathrm{i}}=\mathrm{C}_{\mathrm{j}}=0$ for all $\mathrm{i}, \mathrm{j}$, is | (1) | 1 |
| (Q) | The number of symmetric matrices $\mathrm{M}=\left(\mathrm{a}_{\mathrm{ij}}\right)_{3 \times 3}$ with all entries in T such that $\mathrm{C}_{\mathrm{j}}=0$ for all j , is | (2) | 12 |
| (R) | Let $\mathrm{M}=\left(\mathrm{a}_{\mathrm{ij}}\right)_{3 \times 3}$ be a skew symmetric matrix such that $\mathrm{a}_{\mathrm{ij}} \in \mathrm{T}$ for $\mathrm{i}>\mathrm{j}$. Then the number of elements in the set $\left\{\left(\begin{array}{l}x \ y \ z\end{array}\right): x, y, z \in \mathbb{R}, M\left(\begin{array}{l}x \ y \ z\end{array}\right)=\left(\begin{array}{l}a_{12} \ 0 \ -a_{23}\end{array}\right)\right\}$ is | (3) | infinite |
| (S) | Let $\mathrm{M}=\left(\mathrm{a}_{\mathrm{ij}}\right)_{3 \times 3}$ be a matrix with all entries in T such that $\mathrm{R}_{\mathrm{i}}=0$ for all i . Then the absolute value of the determinant of M is | (4) | 6 |
| | | (5) | 0 |

The correct options is

Accepted Answer

(C) $(\mathrm{P}) \rightarrow(2)(\mathrm{Q}) \rightarrow(4) \quad(\mathrm{R}) \rightarrow(3)(\mathrm{S}) \rightarrow(5)$

Answer

(A) $(\mathrm{P}) \rightarrow(4)(\mathrm{Q}) \rightarrow(2) \quad(\mathrm{R}) \rightarrow(5)(\mathrm{S}) \rightarrow(1)$

Answer

(B) $(\mathrm{P}) \rightarrow(2)(\mathrm{Q}) \rightarrow(4) \quad(\mathrm{R}) \rightarrow(1)(\mathrm{S}) \rightarrow(5)$

Answer

(D) $(\mathrm{P}) \rightarrow(1)(\mathrm{Q}) \rightarrow(5) \quad(\mathrm{R}) \rightarrow(3)(\mathrm{S}) \rightarrow(4)$

JEE Advanced · 2024 · 26 May 2024 · Paper 1 · Matrices

Solution

More JEE Advanced questions on Matrices

List-I		List-II
(P)	The number of matrices $M = (a_{ij})_{3 \times 3}$ with all entries in T such that $R_{i} = C_{j} = 0$ for all $i, j$ , is	(1)	1
(Q)	The number of symmetric matrices $M = (a_{ij})_{3 \times 3}$ with all entries in T such that $C_{j} = 0$ for all j , is	(2)	12
(R)	Let $M = (a_{ij})_{3 \times 3}$ be a skew symmetric matrix such that $a_{ij} \in T$ for $i > j$ . Then the number of elements in the set $⎩ ⎨ ⎧ x y z : x, y, z \in R, M x y z = a_{12} 0 - a_{23} ⎭ ⎬ ⎫$ is	(3)	infinite
(S)	Let $M = (a_{ij})_{3 \times 3}$ be a matrix with all entries in T such that $R_{i} = 0$ for all i . Then the absolute value of the determinant of M is	(4)	6
		(5)	0