What is the largest eigenvalue of the following matrix?

Requested by @Federico Poloni:

Let $A$ be a matrix with positive entries, then from the Perron-Frobenius theorem it follows that the dominant eigenvalue (i.e. the largest one) is bounded between the lowest sum of a row and the biggest sum of a row. Since in this case both are equal to $21$, so must the eigenvalue.

In short: since the matrix has positive entries and all rows sum to $21$, the largest eigenvalue must be $21$ too.

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.

If you sum the row, all the rows they to the same number (21).

That indicates that $\begin {bmatrix} 1\\1\\1 \end{bmatrix}$ must be an eigenvector and 21 is the associated eigenvalue.

The trace of the matrix equal 21, and the sum of the eigenvalues equals the trace.

The remaining two eigenvalues are the negative of one another.

And $3969 < 21^3$ so the other absolute value of the other two eigenvalues are each less than $21$