Exercise 1.5.5

If $Q$ is orthogonal, then there are $n$ bases for matrix $Q$, and they are all independent of each other because of the orthogonality property. So the rank of $Q$ is $n$, which says it is invertible. Since $Q^{-1}=Q^T$, we have $Q{Q}^{-1}=Q{Q}^T=I$, which says the columns of $Q^T$ i.e., the rows of $Q$ are orthogonal, rows of $Q$ are actually columns of $Q^{-1}$, so we know $Q^{-1}$ is also orthogonal.

If $Q_1^T=Q_1^{-1}$ and $Q_2^T=Q_2^{-1}$, we have

So $Q_1{Q}_2$ is an orthogonal matrix.