## The last eigenvalue

The matrix

$$\begin{pmatrix} 10 & 0 & 7 & -7 \\ 4 & 5 & 2 & -2 \\ 16 & 4 & 15 & -8 \\ 30 & 4 & 26 & -19 \\ \end{pmatrix}$$

has three eigenvalues $$3, -4$$ and $$5$$. Determine the remaining eigenvalue.

• #### Resolution

The most tedious way: factorize the characteristic polynomial by the linear polynomials corresponding to the known eigenvalues.

A bit simpler, but still complicated way: use the fact that the product of all eigenvalues is the determinant of the matrix. (Could be derived by substititing 0 to the characteristic polynomial.)

The most simple approach: use the fact that the sum of all eigenvalues is equal to the sum of the elements on the diagonal. (Could be derived from the coefficient by $$t^{n-1}$$ in the characteristic polynomial.)

• #### Result

The missing eigenvalue is 7.