Let P be an odd prime number and `T_(p)` be the following set of `2xx2` matrices :
`T_(P)={A=[(a,b),(c,a)]: a, b, c in {0, 1, ... , p-1}}`
The number of A in `T_(P)` such that det (A) is not divisible by p is

Options

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

Answer

D

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 det (A) is divisible by `p rArr` either `a+b` divisible by `p`
corresponding number of ways `= (p -1)` [excluding zero] or
`(a-b)` is divisible by `p` corresponding number of ways `= p` Total Number of ways ` = 2p -1`