Singular Values & the SVD · Superchallenge problem

Let $L$ be the $10\times 10$ lower-triangular matrix of ones ($\ell_{ij} = 1$ for $j \le i$, else $0$), with singular values $\sigma_1, \ldots, \sigma_{10}$. Compute $\displaystyle\sum_{k=1}^{10}\left(\sigma_k^2 + \frac{1}{\sigma_k^2}\right)$.