Paragraph: Let p be an odd prime number and T_p be the following set of 2 2 matrices T_p= A= [ array ll a & b c & a array ] ; a, b, c 0,1,2, , p-1 Question: 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

Step-by-step solution

Given, A= [ array ll a & b c & a array ] , a, b, c 0,1,2, , p-1 If A is skew-symmetric matrix, then a=0, b=-c |A|=-b^2 . Thus, P divides |A| only when b=0 ...(i) Again, if A is symmetric matrix, then b=c and |A|=a^2-b^2 . Thus, p divides |A| if either p divides (a-b) or p divides (a+b) . p divides (a-b) , only when a=b ie, a=b 0,1,2, ,(p-1) ie, pchoices p divides (a+b) . p choices, including a=b=0 included in (i) Total number of choices are (p+p-1)=2 p-1 . Hence, (c) is the correct option.