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Let S be the set containing all 3 x 3 matrices with ent... | null (Mathematics) | DivineJEE

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Mathematics

Exam

Mathematics

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 $A^{\mathrm{T}} A$ is 6 is ____________.

Correct Answer:

5376

Explanation

Sum of all diagonal elements is equal to sum of square of each element of the matrix.

i.e.,

A = \left[ {\matrix{ {{a_1}} & {{a_2}} & {{a_3}} \cr {{b_1}} & {{b_2}} & {{b_3}} \cr {{c_1}} & {{c_2}} & {{c_3}} \cr } } \right]

then

{t_r}\,(A\,.\,{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

\because

{a_i},{b_i},{c_i} \in \{ - 1,0,1\}

for

i = 1,2,3

\therefore

Exactly three of them are zero and rest are 1 or

-

1.

Total number of possible matrices

{}^9{C_3} \times {2^6}

= {{9 \times 8 \times 7} \over 6} \times 64

= 5376

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