- #1

ArcanaNoir

- 779

- 4

## Homework Statement

If P is an orthogonal matrix with detP = -1, show that I+P has no inverse. (Hint: show that (P^t)(I+P)=(I+P)^t)

P^t is P transposed.

I is the identity matrix given by PP^t=I

a^-1 means inverse a

a, b, P and such letters, capital or otherwise, are all matrices, limit to square matrices for our purposes here.

## Homework Equations

If I knew all the relevant equations I wouldn't be stuck...

For the transpose, I know (ab)^t=(b^t)(a^t) , (a+b)^t= a^t + b^t , (a^t)^-1 = (a^-1)^t

and A= .5(A + A^t) + .5(A - A^t). (maybe this is the important one?)

I know for orthogonal matrices P^t=P^-1 and P(P^t)=I

For determinates, I know det(I)=1 , det(A^-1)=1/(detA) , det(AB)=det(A)det(B) , det(A^t)=det(A) and the determinate of orthogonal matrices is +1 or -1, called proper when positive and improper when negative.

Also, the rank of an invertible matrix must equal the number of rows (or columns, same thing), and the determinate mustn't be 0.

## The Attempt at a Solution

Now, to complicate matters, we have not discussed orthogonal matrices and some relevant related topics in class due to schedule limitations and guest high school students. This question is part of a list of similar questions for a special project only for the undergrads in the class. Everything I know about orthogonal matrices I read out of an assortment of textbooks in the library. I successfully completed other proofs using simply structured equations like the ones above and I would like to solve this problem similarly, leaving out the complicated bits about orthonormal columns and something or others that I haven't quite understood. If it cannot be done simply, then I am willing to delve deeper into that stuff. I do want, however, to avoid geometric arguments and explanations like the plague. I am not a visual thinker, and have zero basis for understanding linear algebra in this way, so to try to do so now would not be the best approach.

So, I'm not looking for someone to just give me the proof, I'm braver than that. I just don't know what I'm not seeing. Surely I'm missing some equation or relationship here. I feel like the determinate is important but I don't know any applicable additive properties of determinates or ranks or anything. Replacing I with PP^t got me no further. It seems interesting that detI = 1 and detP = -1 which when added together = 0 and a determinate of zero gives a noninvertible matrix, but I know you can't just go around adding determinates... Right?