Exercise 8

If $Q$ is an orthogonal matrix, remember that $Q^TQ=I$. Explain why $\det Q = \pm 1$.

If $A=U\Sigma V^T$ is a singular value decomposition of a square matrix $A$, explain why $|\det A|$ is the product of the singular values of $A$.

What does this say about the singular values of $A$ if $A$ is invertible?