The last eigenvalue

Task number: 2661

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.

Difficulty level: Moderate task
Proving or derivation task
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