Question

Let p be an odd prime number and `T_p` be the following set of 2 x 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) is divisible by p is: [Note: the trace of a matrix is the sum of its diagonal entries.]
A. `(p-1)^(2)`
B. `2(p-1)`
C. `(p-1)^(2)+1`
D. `2p-1`

Answer 1

Correct Answer - D
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`