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  4. 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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Q. Let $S$ be the set containing all $3 \times 3$ matrices with entries from $\{-1,0,1\}$. The total number of matrices A $\in S$ such that the sum of all the diagonal elements of $A ^{ T } A$ is 6 is _____

JEE MainJEE Main 2022Determinants

Solution:

$ \operatorname{Tr}\left(A A^T\right)=6$
$ AA ^{ T }= \begin{bmatrix}a & d & g \\b & e & h \\c & f & i\end{bmatrix}\begin{bmatrix} a & b & c \\d & e & f \\g & h & i\end{bmatrix}$
Now give
$ a^2+d^2+g^2+b^2+e^2+h^2+c^2+f^2+i^2=6 $
$ ={ }^9 C_3 \times 2^6$
$ =5376$