The elements of $3 \times 3$ matrix $A$ are either $1$ or $-1$, then
Step-by-Step Solution
Key Concept: For a 3×3 matrix with entries ±1, the determinant |A| is an even integer bounded by properties of symmetric matrices. The maximum |det(A)| for symmetric matrices with ±1 entries is 4, achieved through careful arrangement of elements while maintaining a₁₁ = a₂₂ = a₃₃ and aᵢⱼ = aⱼᵢ constraints.
For a symmetric matrix $A$ with elements $a, b, c, d, e, f$, there are $2^6 = 64$ possible matrices. The maximum value of $|A|$ is found to be 4, achieved when the matrix takes the form $\begin{pmatrix} 1 & -1 & 1 \\ -1 & 1 & 1 \\ -1 & -1 & 1 \end{pmatrix}$. By interchanging any two rows or columns, the minimum value of $|A|$ is $-4$, and the maximum value of trace is 3.
Correct Answer: [A-s] [B-q] [C-r] [D-p]