Let `p` be an odd prime number and `T_(P)` be the following
set of `2xx2` matrices.
`T_(p)={A= [[a,b],[c,d]], 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. `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)`