CAIE
FP1
2014
June
Q9
10 marks
Standard +0.3
9 The matrix \(\mathbf { M }\), where
$$\mathbf { M } = \left( \begin{array} { r r r }
- 2 & 2 & 2 \\
2 & 1 & 2 \\
- 3 & - 6 & - 7
\end{array} \right)$$
has an eigenvector \(\left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right)\). Find the corresponding eigenvalue.
It is given that if the eigenvalues of a general \(3 \times 3\) matrix \(\mathbf { A }\), where
$$\mathbf { A } = \left( \begin{array} { l l l }
a & b & c \\
d & e & f \\
g & h & i
\end{array} \right)$$
are \(\lambda _ { 1 } , \lambda _ { 2 }\) and \(\lambda _ { 3 }\) then
$$\lambda _ { 1 } + \lambda _ { 2 } + \lambda _ { 3 } = a + e + i$$
and the determinant of \(\mathbf { A }\) has the value \(\lambda _ { 1 } \lambda _ { 2 } \lambda _ { 3 }\).
Use these results to find the other two eigenvalues of the matrix \(\mathbf { M }\), and find corresponding eigenvectors.