Suppose that we wish to compute the inverse and log-determinant of the matrix

\[ \mathbf{I}_{n}+\mathbf{U} \mathbf{U}^{\top} \]

where \(\mathrm{U}\) is an \(n \times h\) matrix with \(h \ll n\). Show that

\[ \left(\mathbf{I}_{n}+\mathbf{U} \mathbf{U}^{\top}\right)^{-1}=\mathbf{I}_{n}-\mathbf{Q}_{n} \mathbf{Q}_{n}^{\top} \]
where \(\mathbf{Q}_{n}\) contains the first \(n\) rows of the \((n+h) \times h\) matrix \(\mathbf{Q}\) in the \(\mathbf{Q R}\) factorization of the \((n+h) \times h\) matrix:
\[ \left[\begin{array}{c} \mathbf{U} \\ \mathbf{I} \end{array}\right]=\mathbf{Q R} \]
In addition, show that \(\ln \left|\mathbf{I}_{n}+\mathbf{U} \mathbf{U}^{\top}\right|=\sum_{i=1}^{h} \ln r_{i i}^{2}\), where \(\left\{r_{i i}\right\}\) are the diagonal elements of the \(h \times h\) matrix \(\mathbf{R}\).