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 x 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) is divisible by p is: [Note: the trace of a matrix is the sum of its diagonal entries.]A. `(p-1)^(2)`B. `2(p-1)`C. `(p-1)^(2)+1`D. `2p-1` |
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Answer» Correct Answer - D 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 det (A) is divisible by `p rArr` either `a+b` divisible by `p` corresponding number of ways `= (p -1)` [excluding zero] or `(a-b)` is divisible by `p` corresponding number of ways `= p` Total Number of ways ` = 2p -1` |
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