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 ______.

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TestHub · Accepted Answer

Given that the sum of all the diagonal elements of $A^{T} A$ is $6$ , means sum of all diagonal elements is equal to sum of square each element of the matrix.
i.e., $A = [\begin{array}{l} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array}]$ and $A^{T} = [\begin{array}{l} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array}]$
Then $t_{r} (A \cdot A^{T})$ $= a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + b_{1}^{2} + b_{2}^{2} + b_{3}^{2} + c_{1}^{2} + c_{2}^{2} + c_{3}^{2}$
Given $a_{i}, b_{i}, c_{i} \in \{- 1, 0, 1\}$ for $i = 1, 2, 3$
Now sum to be $6$ exactly three of them should be zero and rest should be $1$ or $- 1$ .
So, total number of possible matrices ${}^{9}C_{3} \times 2^{6}$
$= \frac{9 \times 8 \times 7}{6} \times 64$
$= 5376$

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