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.]

Options

A. `2P^(2)`
B. `p^(3)-5p`
C. `p^(3) 3p`
D. `P^(3) = p^(2)`

Answer

B

Explanation

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`
Total number of `A= pxx pxxp=p^(3)`
Number of A such that det `(A)` divisible by `p`
`= (p-1)^(2)+` numbre of A in which `a = 0`
`= (p-1)^(2)+ p + p -1 =p^(2)`
Required number `= p^(3) - p^(2)`