Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
A be the set of all square matrices of order 3 with elements either 0, 1, or $-1$, then:
$O(A) = 3^9$
Number of symmetric matrices in A whose trace is 0 = 7 × Number of skew symmetric matrices in A
Number of matrices in A such that each of 0, 1, and $-1$ occurs atleast once at any position is 18150
All skew symmetric matrices in A are singular
Step-by-Step Solution
Key Concept: Use combinatorial counting with inclusion-exclusion principle to count matrices satisfying trace and symmetry constraints.
For each of 3 positions in a $3×3$ matrix, there are 3 possibilities (entries from $\{0, 1, -1\}$), giving $O(A) = 3^9$ total matrices. Matrices with trace zero number $3^9 - C_1^9·2^9 - C_2^9·2^9 = 19683 - 3×512 + 3 = 18150$. Skew-symmetric matrices of odd order have determinant zero. The count includes $7×3^3$ matrices with trace zero and $3^3$ skew-symmetric matrices, yielding total matrices where each of $0, 1, -1$ occurs at least once.
Correct Answer: 1,2,3,4