InterviewSolution
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InterviewSolution
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Let p be an odd prime number and `T_p`, be the following set of `2 xx 2` matrices `T_p={A=[(a,b),(c,a)]:a,b,c in {0,1,2,.........p-1}}` The number of A in `T_p`, such that A is either symmetric or skew-symmetric or both, and det (A) divisible by p isA. `(p-1)(p^(2)-p+1)`B. `p^(3)-(p-1)^(2)`C. `(p-1)^(2)`D. `(p-1) (p^(2) - 2)` |
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Answer» Correct Answer - A If A is symmetric matrix, then b = c `therefore det (A) = abs((a,b),(b,a))= a^(2) - b^(2) = (a+b) (a-b)` `a, b, c, in {0, 1, 2, 3,..., P-1}` Number of numbers of type `np=1` `np+1=1` `np + 2 =1` ` ………` `……….` `np_(p-1) = 1 AA n in I` As Tr (A) not divisible by `p rArr a ne 0 ` det `(A)` is divisible by `p rArr a^(2) - bc` divisible by `p` Number of ways of selection of a, b, c `= (p-1) [(p-1) xx1] = (p-1)^(2)` |
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