Normalized eigenvectors

4. $$\mathbf { M } = \left( \begin{array} { l l l } 6 & k & 2 \\ k & 5 & 0 \\ 2 & 0 & 7 \end{array} \right)$$ where \(k\) is a constant. Given that 3 is an eigenvalue of \(\mathbf { M }\),

1. determine the possible values of \(k\). Given that \(k < 0\)
2. determine the other eigenvalues of \(\mathbf { M }\).
3. Determine a normalised eigenvector corresponding to the eigenvalue 3

4. $$\mathbf { M } = \left( \begin{array} { l l l } 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{array} \right)$$

1. Show that 6 is an eigenvalue of the matrix \(\mathbf { M }\) and find the other two eigenvalues of \(\mathbf { M }\).
2. Find a normalised eigenvector corresponding to the eigenvalue 6

3. $$\mathbf { M } = \left( \begin{array} { c c c } 3 & - 4 & k \\ 1 & - 2 & k \\ 1 & - 5 & 5 \end{array} \right) \text { where } k \text { is a constant }$$ Given that 3 is an eigenvalue of \(\mathbf { M }\),

1. find the value of \(k\).
2. Hence find the other two eigenvalues of \(\mathbf { M }\).
3. Find a normalised eigenvector corresponding to the eigenvalue 3
   .

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