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5 years ago Latika Leekha
165 Points
```							Hello student,The passage saysp is an odd prime number and TP denotes such a set of 2X2 matrices whose elements are of the form a, b in first row and c, a in second row where a, b, c all elements are from the set {0, 1, 2, …. , p-1). this means that the elements of the matrix can be numbers ranging from 0 till 1 less than that odd prime number p. Infact all prime numbers gretaer than 2 are odd.Further on the basis of this you are required to answer the following questions.I am solving the following question too:The number of A in TP such that A is either symmetric or skew-symmetric or both and det is also divisible by p is 2p-1.For symmetric , we will have b = c and so det = a2-b2Similarly, for skew-symmetric, a = 0 and b = -c and so det = b2.Now for det to be divisible by p, in symmetric we have a2-b2 = (a-b)(a+b) should be divisible by p i.e. = kpNow for this to happen, either (a-b) or (a+b) shoulod be divisible by p.(a-b) is divisible only when a-b = 0 i.e. a = bThere are p possibilities in this case.Next, (a+b) can be divisible when a+b = 0.pi.e. a and b are negatives – not possible.Also, if a = 0 then b = p not possible.LEaving these cases there are a total of (p-1) possibilities.Hence, total nuber of possibilities are p + p - 1 = 2p -1.Now, second case i.e. skew-symmetric, det = b2, so b2 should be divisible by pi.e. b.b should be divisible by p i .e.b should be divisible by p.but then b = 0 and a = 0 because it is skew-symmetric.But this case has already been included in the previous case.Hence, toal number of cases 2p-1.
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5 years ago
```							thank you it was very helpful...
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5 years ago
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