Let `S` be the set of `2xx2` matrices given by `S={A=[[a,b],[c,d]],"where" a,b,c,d,in I}`, such that `A^(T)=A^(-1)` Then

Options

A. number of matrices in set s is equal to 6
B. number of matrices in set S such that `|A-I_(2)| ne 0` is equal to 3
C. symmetric matrices are more than skew -symmetric matrices in set S
D. all matrices in set S are singular

Answer

B, C

Explanation

As, `A A^(T)=I_(2)`
`implies [[a,b],[c,d]],[[a,c],[b,d]]=[[1,0],[0,1]]`
`impliesa=0,b=pm1,d=0,c=pm1`
therefore Total 8 matrices are possible They are
`[[1,0],[0,1]],[[1,0],[0,-1]],[[-1,0],[0,1]],[[-1,0],[0,-1]]`
`[[0,1],[1,0]],[[0,-1],[1,0]],[[0,1],[-1,0]],[[0,-1],[-1,0]]`
Also `|A-I_(2)|=|A-A A^(T)|=|A||I_(2)-A^(T)|`
`=|A||(I_(2)-A^(T))^(T)|=|A||I_(2)-A|`
`=|A||A-I_(2)|`
`implies|A|=1(As,|A-I_(2)|ne 0)`
except `A=1=[[1,0],[0,1]]`
where `|A|=1 but`
`| A-I_(2)|=0`