Question

Let S be the set containing all 3 × 3 matrices with entries from {-1, 0, 1}. The total number of matrices A ∈ S such that the sum of all the diagonal elements of A^T A is 6 is ________.

Answer 1

Correct Answer: 5376

The correct answer is 5376
Sum of all diagonal elements is equal to sum of square of each element of the matrix.
\(i.e., A =\) \(\begin{bmatrix}   a_1 & a_2 & a_3 \\   b_1 & b_2 & b_3 \\   c_1 & c_2 & c_3 \\ \end{bmatrix}\)
then \(t_r(A.A^T)\)
\(= a^{2}_{1}+a^{2}_{2}+a^{2}_{3}+b^{2}_{1}+b^{2}_{2}+b^{2}_{3}+c^{2}_{1}+c^{2}_{2}+c^{2}_{3}\)
\(∵ a_i, b_i, c_i ∈{-1,0,1} \)for \(i = 1,2,3\)
∴ Exactly three of them are zero and rest are 1 or – 1.
Total number of possible matrices
\(^9C_3×2^6\)
\(= \frac{9×8×7}{6}×64\)
= 5376