Question

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

Answer 1

Correct Answer - D
We must have `a^(2)-b^(2)=kp`
`implies (a+b) (a-b)=kp`
`implies` either `a-b=0` or `a+b` is multiple of p when a=b, number of matrices is p
and when `a+b=` multiple of `p implies a, b` has `p-1`
`:.` total number of matrices `=p+p-1=2p-1`.