A be set of 3 × 3 matrices formed by entries 0 , − 1 , and 1 only. Also each of 1 , − 1 , 0 occurs exactly three times in each matrix. The number of symmetric matrices with trace ( A ) = 0 is k , then

Question

A be set of 3 × 3 matrices formed by entries 0 , − 1 , and 1 only. Also each of 1 , − 1 , 0 occurs exactly three times in each matrix. The number of symmetric matrices with trace ( A ) = 0 is k , then 6 k ​ = ……… .

TestHub · Accepted Answer

For non-diagonal entries, we required even no. of 1 , even no. of -1 and even no. of 0 , for diagonal three entries are remained, − 1 , 0 , 1 . So no. of cases in which trace = 0 are 3 ! And no. of symmetric matrices for each arrangement of 1 , − 1 , 0 in diagonal = 3 ! Total such matrices = 3 ! × 3 ! = 36

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