IfUUT I and U is a square matrix prove quoting theorems tha

IfUU^T = I and U is a square matrix, prove (quoting theorems) that detU = ±1

Solution

Theorem

The determinant of the product of two square matrices is the product of their determinants, that is,

det(AB)=det(A)det(B) where A and B are square matrices

So,

det(UU^T)=det(U)det(U^T)

Theorem

Determinant of a transpose of a square matrix A is equal to the determinant of the matrix A

SO,

det(UU^T)=det(U)det(U^T)=det(U)^2

But UU^T=I

So,

det(UU^T)=det(U)^2=det(I)=1

Hence,

det(U)=+-1