Projection Linear Transformations

Question

Let $T: ℝ^2 \rightarrow ℝ^2$ be a projection onto the line $y = 3x.$

What would the determinant of the corresponding matrix (let's call it $A$) be?

Answer 1

This projection operator $T$ is not invertible, and hence det$A=0.$

A quick argument: observe that $T((3, -1))=0$ ie the kernel/nullspace of $T$ is not trivial. Hence, $T$ is not invertible.

Now, it's your job to show that in general projections (except, identity) are not invertible.