I'm currently doing an old exam (linear algebra), my real one is in three days. There was a question in it and I wanted to make sure that my way of thinking is correct. The question is:
When one expresses the following maps as matrix; in how many cases are the matrices non-singular?
- a rotation in $\mathbb{R}^2$,
- an orthogonal projection in $\mathbb{R}^2$ onto a line,
- an oblique projection in $\mathbb{R}^2$ onto a line,
- the reflection in $\mathbb{R}^2$ in the origin.
My answer:
- A rotation matrix is non-singular since it is invertible.
- Now I know that all projection matrices except the identity matrix are singular. The identity matrix is an orthogonal projection since: $I^2 = I$ and $I^T = I$. This implies that an orthogonal projection can be non-singular in the case that the projection matrix is the identity matrix.
- A projection matrix $P$ is oblique $\iff$ $P^2 = P$ and $P \ne P^T$. I figured that, since the only matrix that is a projection and is non-singular is $I$, and since $I = I^T \implies \not\exists P : P^2 = P$ and $P^T \ne P$.
- A reflection is invertible and hence non-singular.
Which concludes that in three cases the matrix can be non-singular.
I'm very curious whether or not I'm missing something and I would appreciate if someone could point out is that is the case. When doing the homework problems I have never encountered a question like this before. Thank you.
Edit: It is a multiple choice question where I can choose between 1,2,3,4 and 5 possibilities where a matrix can be non-singular. So I don't have to prove it, however, im curious if my way of thinking is correct. (Why there is a fifth option in the answer I don't know)
Answering my own question.
At option number 2 in the question, that ask if the matrix for an orthogonal projection in $\mathbb{R}^2$ onto a line is non-singular. Indeed, the identity matrix is a projection, however not onto a line. This rules out the case of the identity matrix and hence no such matrix can be invertible. Concluding that there are two cases in which the matrix is non-singular, namely, the rotation and the reflection.