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)(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)`