Homework help! Matrix determinant question?

[svalik]

What is wrong with the proof that projection matrices have det (P) = 1?

P = A(AtA)^-1At

so

|P| = |A| (1/ (|At| |A|)) |At| = 1


I understand that that det (AB) = |A| |B|, so what's wrong with this proof?

 

[Unco]

\(\displaystyle P = A(A^{T}A)^{-1}A^{T}\)

Is A a square matrix? If not, then det(A) is not defined. For your proof, consider the property of projection matrices that \(\displaystyle P^2=P\).

 

[svalik]

Yes I considered that if A isn't a square matrix then it doesn't have a determinant. But how do I find out whether A is a square matrix?

 

[Unco]

You can still show that P^2= P regardless of the dimensions of A.