What is the largest eigenvalue of the following matrix? 
Find the largest eigenvalue of the following matrix
  $$\begin{bmatrix}
 1 &  4 & 16\\
 4 & 16 &  1\\
16 &  1 &  4
\end{bmatrix}$$

This matrix is symmetric and, thus, the eigenvalues are real. I solved for the possible eigenvalues and, fortunately, I found that the answer is $21$.
My approach:
The determinant on simplification leads to the following third degree polynomial.
$$\begin{vmatrix}
1-\lambda & 4 &16\\
4 &16-\lambda&1\\
16&1&4-\lambda
\end{vmatrix}
= \lambda^3-21\lambda^2-189\lambda+3969.$$
At a first glance seen how many people find the roots of this polynomial  with pen and paper using elementary algebra. I managed to find the roots and they are $21$, $\sqrt{189}$, and $-\sqrt{189}$ and the largest value is $21$.
Now the problem is that my professor stared at this matrix for a few seconds and said that the largest eigenvalue is $21$. Obviously, he hadn't gone through all these steps to find that answer. So what enabled him answer this in a few seconds? Please don't say that he already knew the answer.
Is there any easy way to find the answer in a few seconds? What property of this matrix makes it easy to compute that answer?
Thanks in advance.

The trick is that $\frac1{21}$ of your matrix is a doubly stochastic matrix with positive entries, hence the bound of 21 for the largest eigenvalue is a straightforward consequence of the Perron-Frobenius theorem.
