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

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