Let p be an odd prime number and `T_p`, be the following set of `2 xx 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) divisible by p is

Options

A. `(p-1)(p^(2)-p+1)`
B. `p^(3)-(p-1)^(2)`
C. `(p-1)^(2)`
D. `(p-1) (p^(2) - 2)`

Answer

A

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`
As Tr (A) not divisible by `p rArr a ne 0 `
det `(A)` is divisible by `p rArr a^(2) - bc` divisible by `p`
Number of ways of selection of a, b, c
`= (p-1) [(p-1) xx1] = (p-1)^(2)`