UPSC NDA Math Model Paper 7

Given: AAT=I=AT A where A is a square matrix and I is the identity matrix.
Here, we have to find |A|
∵AAT=I
⇒|A.AT|=|I|=1 - - - - - - - -(∵Determinant of an identity matrix is always 1)
As we know that, if A and B are two determinants of order n, then |A.B|=|A|.|B|
⇒|A.AT|=|A|.|AT|=1
As we know that, for any square matrix say A,|A|=|AT|
⇒|A|2=1
⇒|A|=±1
Hence, option B is the correct answer.