Question

find the eigenvalue and eigenvector for matrix


0 0 5
S= -2 1 1
3 0 2

Answer 1

Conditions

find the eigenvalue and eigenvector for matrix

$$S = \begin{array}{ccc} & 0 & 0 \\ -2 & 1 & 1 \\ 3 & 0 & 2 \end{array}$$

Please show your work

Solution

$$S = \left( \begin{array}{ccc} 0 & 0 & 5 \\ -2 & 1 & 1 \\ 3 & 0 & 2 \end{array} \right)$$

The definition of an eigenvalue claims, that they are values of $\lambda$, which could be found by solving the following matrix equation:

$$\begin{aligned} \det(S - \lambda E) &= 0 \\ |S - \lambda E| &= \left| \begin{array}{ccc} - \lambda & 0 & 5 \\ -2 & 1 - \lambda & 1 \\ 3 & 0 & 2 - \lambda \end{array} \right| = - \lambda \left| \begin{array}{cc} 1 - \lambda & 1 \\ 0 & 2 - \lambda \end{array} \right| - 0 + 5 \left| \begin{array}{cc} -2 & 1 - \lambda \\ 3 & 0 \end{array} \right| \\ &= - \lambda (1 - \lambda) (2 - \lambda) - 0 - 0 + 0 - 15(1 - \lambda) = ((- \lambda) (2 - \lambda) - 15)(1 - \lambda) \\ &= 0 \end{aligned}$$ $$\begin{aligned} \lambda_1 &= 1 \\ (-\lambda)(2 - \lambda) - 15 &= 0 \\ \lambda^2 - 2\lambda - 15 &= 0 \\ \lambda_2 &= -3 \\ \lambda_3 &= 5 \end{aligned}$$

Answer: The eigenvalues are:

$$\begin{aligned} \lambda_1 &= 1 \\ \lambda_2 &= -3 \\ \lambda_3 &= 5 \end{aligned}$$