In Prof. Strang's book exercise 29 of the section on determinants (5.1) asks what is wrong with the following proof that all projection matrixes have determinant 1.

(using ' for transpose)
|P| = | A inv(A'A) A' | = |A| (1/|A'||A|) |A'| = |A||A'| / |A'||A| = 1

I thought it had someting to do with the order one takes the inverse and the determinant, but now I'm not so sure anymore. Anyone knows what's wrong with that proof? Thanks.

Question

In Prof. Strang's book exercise 29 of the section on determinants (5.1) asks what is wrong with the following proof that all projection matrixes have determinant 1.

(using ' for transpose)
|P| = | A inv(A'A) A' | = |A| (1/|A'||A|) |A'| = |A||A'| / |A'||A| = 1

I thought it had someting to do with the order one takes the inverse and the determinant, but now I'm not so sure anymore. Anyone knows what's wrong with that proof? Thanks.

anonymous · Answer

It's that A may not be invertible, hence its determinant (and that of A transposed) can be zero.