Let \alpha and \beta be the distinct roots of the equation x^2+x-1=0. Consider the set T=\{1, \alpha, \beta\}. For a 3 imes 3 matrix M= (a_{i j}) 3 imes 3), define R_i=a_{i 1}+a_{i 2}+a_\beta and C_j=a_{1 j}+a_{2 j}+a_{3 j} 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 List-II (P) The number of matrices M= (a_{i j}) 3 imes 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_{i j}) 3 imes 3 with all entries in T such that C_j=0 for all j is (2) 12 (R) Let M= (a_{i j}) _{3 imes 3} be a skew symmetric matrix such that a_{ij} \in T for ij . Then the number of elements in the set \{\left(\begin{array}{c}x \ y \ z)\end{array}: x, y \cdot z \in R, M\left(\begin{array}{c}x \ y \ z)\end{array}=\left(\begin{array}{c}a_{12\end{array} ; 0 ; -a_{23})}\} is (3) Infinite (S) Let M= (a_{i j}) _{3 imes 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 The correct option is
[JEE Adv 2024 P1]

Question

Let \alpha and \beta be the distinct roots of the equation x^2+x-1=0. Consider the set T=\{1, \alpha, \beta\}. For a 3 	imes 3 matrix M= (a_{i j}) 3 	imes 3), define R_i=a_{i 1}+a_{i 2}+a_\beta and C_j=a_{1 j}+a_{2 j}+a_{3 j} 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 List-II (P) The number of matrices M= (a_{i j}) 3 	imes 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_{i j}) 3 	imes 3 with all entries in T such that C_j=0 for all j is (2) 12 (R) Let M= (a_{i j}) _{3 	imes 3} be a skew symmetric matrix such that a_{ij} \in T for i>j . Then the number of elements in the set \{\left(\begin{array}{c}x \ y \ z)\end{array}: x, y \cdot z \in R, M\left(\begin{array}{c}x \ y \ z)\end{array}=\left(\begin{array}{c}a_{12\end{array} ; 0 ; -a_{23})}\} is (3) Infinite (S) Let M= (a_{i j}) _{3 	imes 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 The correct option is
[JEE Adv 2024 P1]

Examsnet · Accepted Answer

Correct Answer is : (P) ⟶ (2) (Q) ⟶ (4) (R) ⟶ (3) (S) ⟶ (5)

Matrices and Determinants