Question 16.8.2: Find the values of λ for which the following system of equat......

The variables x, y, and z appear on both sides of the equations, so we start by putting the system into standard form:

According to Theorem 16.8.2, this system has a nontrivial solution if and only if the coefficient matrix is singular:

The value of the determinant is found to be λ(1 − λ)(λ − 6). Hence, system (∗) has nontrivial solutions if and only if λ = 0, 1, or 6.^{10}

^{10} Using terminology explained in FMEA, this example asks us to find the eigenvalues of the coefficient matrix \begin{pmatrix} 5 & 2 &1 \\ 2 & 1 &0 \\1 & 0 &1 \end{pmatrix}