The correct options are
A Number of matrices in set S is equal to 8.
C Symmetric matrices are more than skew symmetric matrices in set S.
AT=A−1
AAT=I
[abcd][acbd]=[1001]
a=0, b=±1, d=0, c=±1 ora=±1, b=0, d=±1, c=0
So total 8 matrices are possible.
[1001],[100−1],[−1001],[−100−1],[0110],[0−110],[01−10],[0−1−10]
|A−I2|=|A−AAT|=|A||I2−AT|=|A||(I2−AT)T|=|A||I2−A|
|A−I2|=|A||I2−A|
As |A−I2|≠0⇒|A|=1
So total 3 such matrices are possible
except 1 case
A=I=[1001] where |A−I2|=0