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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 det (A) is not divisible by p , is

Options

  1. A2 p^2
  2. Bp^3-5 p
  3. Cp^3-3 p
  4. Dp^3-p^2

Correct answer

D. p^3-p^2

Step-by-step solution

The number of matrices for which p does not divide Tr (A)=(p-1) p^2 of these (p-1)^2 are such that p divides |A| . The number of matrices for which p divides Tr (A) and p does not divides |A| are (p-1)^2 aligned & Required number & =(p-1) p^2-(p-1)^2+(p-1)^2 & =p^3-p^2 aligned

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