InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
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. `2P^(2)`B. `p^(3)-5p`C. `p^(3) 3p`D. `P^(3) = p^(2)` |
|
Answer» Correct Answer - B 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` Total number of `A= pxx pxxp=p^(3)` Number of A such that det `(A)` divisible by `p` `= (p-1)^(2)+` numbre of A in which `a = 0` `= (p-1)^(2)+ p + p -1 =p^(2)` Required number `= p^(3) - p^(2)` |
|