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Let P be an odd prime number and T(p) be...

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

A

`(p-1)^(2)`

B

`2(p-1)`

C

`(p-1)^(2)+1`

D

`2p-1`

Text Solution

Verified by Experts

The correct Answer is:
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`.
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