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Paragraph: Let p be an odd prime number and T_p be the following set of 2 2 matrices T_p= A= [ array ll a & b c & a array ] ; a, b, c 0,1,2, , p-1 Question: The number of A in T_p such that the trace of A is not divisible by p but det (A) is divisible by p is [Note : The trace of a matrix is the sum of its diagonal entries.]

Options

  1. A(p-1) (p^2-p+1 )
  2. Bp^3-(p-1)^2
  3. C(p-1)^2
  4. D(p-1) (p^2-2 )

Correct answer

C. (p-1)^2

Step-by-step solution

Trace of A=2 a , will be divisible by p iff a=0 . |A|=a^2-b c , for (a^2-b c ) to be divisible by p . There are exactly (p-1) ordered pairs (b, c) for any value of a . Required number is (p-1)^2 . Hence, (c) is the correct option.

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