Eigenvalues — Practice Problems and Worked Solutions

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18 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find the distinct eigenvalues.

   $$A=\left[\begin{array}{cc}2& 0\\ 0& 3\end{array}\right]$$

   Answer: $\left\{2,3\right\}$

   Step-by-step solution

   1. $$\det (\lambda I-A)={\lambda }^2-5\lambda +6$$
   2. $$\lambda \in \left\{2,3\right\}$$
2. 2.

   Find the distinct eigenvalues.

   $$A=\left[\begin{array}{cc}1& 1\\ 0& 4\end{array}\right]$$

   Answer: $\left\{1,4\right\}$

   Step-by-step solution

   1. $$\det (\lambda I-A)={\lambda }^2-5\lambda +4$$
   2. $$\lambda \in \left\{1,4\right\}$$
3. 3.

   Find the distinct eigenvalues.

   $$A=\left[\begin{array}{cc}5& 0\\ 2& -1\end{array}\right]$$

   Answer: $\left\{-1,5\right\}$

   Step-by-step solution

   1. $$\det (\lambda I-A)={\lambda }^2-4\lambda -5$$
   2. $$\lambda \in \left\{-1,5\right\}$$
4. 4.

   Find the distinct eigenvalues.

   $$A=\left[\begin{array}{cc}3& 1\\ 0& 3\end{array}\right]$$

   Answer: $\left\{3\right\}$

   Step-by-step solution

   1. $$\det (\lambda I-A)={\lambda }^2-6\lambda +9$$
   2. $$\lambda \in \left\{3\right\}$$
5. 5.

   Find the distinct eigenvalues.

   $$A=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]$$

   Answer: $\left\{1,3\right\}$

   Step-by-step solution

   1. $$\det (\lambda I-A)={\lambda }^2-4\lambda +3$$
   2. $$\lambda \in \left\{1,3\right\}$$
6. 6.

   Find the distinct eigenvalues.

   $$A=\left[\begin{array}{cc}4& 2\\ 1& 3\end{array}\right]$$

   Answer: $\left\{2,5\right\}$

   Step-by-step solution

   1. $$\det (\lambda I-A)={\lambda }^2-7\lambda +10$$
   2. $$\lambda \in \left\{2,5\right\}$$
7. 7.

   Find the distinct eigenvalues.

   $$A=\left[\begin{array}{cc}5& 0\\ 0& 2\end{array}\right]$$

   Answer: $\left\{2,5\right\}$

   Step-by-step solution

   1. $$\det (\lambda I-A)={\lambda }^2-7\lambda +10$$
   2. $$\lambda \in \left\{2,5\right\}$$
8. 8.

   Find the distinct eigenvalues.

   $$A=\left[\begin{array}{cc}1& 0\\ 0& 6\end{array}\right]$$

   Answer: $\left\{1,6\right\}$

   Step-by-step solution

   1. $$\det (\lambda I-A)={\lambda }^2-7\lambda +6$$
   2. $$\lambda \in \left\{1,6\right\}$$
9. 9.

   Find the distinct eigenvalues.

   $$A=\left[\begin{array}{cc}4& 0\\ 1& -2\end{array}\right]$$

   Answer: $\left\{-2,4\right\}$

   Step-by-step solution

   1. $$\det (\lambda I-A)={\lambda }^2-2\lambda -8$$
   2. $$\lambda \in \left\{-2,4\right\}$$
10. 10.

    Find the distinct eigenvalues.

    $$A=\left[\begin{array}{cc}2& 2\\ 0& 2\end{array}\right]$$

    Answer: $\left\{2\right\}$

    Step-by-step solution

    1. $$\det (\lambda I-A)={\lambda }^2-4\lambda +4$$
    2. $$\lambda \in \left\{2\right\}$$
11. 11.

    Find the distinct eigenvalues.

    $$A=\left[\begin{array}{cc}3& 0\\ 5& 1\end{array}\right]$$

    Answer: $\left\{1,3\right\}$

    Step-by-step solution

    1. $$\det (\lambda I-A)={\lambda }^2-4\lambda +3$$
    2. $$\lambda \in \left\{1,3\right\}$$
12. 12.

    Find the distinct eigenvalues.

    $$A=\left[\begin{array}{cc}7& 0\\ 0& 3\end{array}\right]$$

    Answer: $\left\{3,7\right\}$

    Step-by-step solution

    1. $$\det (\lambda I-A)={\lambda }^2-10\lambda +21$$
    2. $$\lambda \in \left\{3,7\right\}$$
13. 13.

    Find the distinct eigenvalues.

    $$A=\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$$

    Answer: $\left\{-1,3\right\}$

    Step-by-step solution

    1. $$\det (\lambda I-A)={\lambda }^2-2\lambda -3$$
    2. $$\lambda \in \left\{-1,3\right\}$$
14. 14.

    Find the distinct eigenvalues.

    $$A=\left[\begin{array}{cc}5& 4\\ 1& 2\end{array}\right]$$

    Answer: $\left\{1,6\right\}$

    Step-by-step solution

    1. $$\det (\lambda I-A)={\lambda }^2-7\lambda +6$$
    2. $$\lambda \in \left\{1,6\right\}$$
15. 15.

    Find the distinct eigenvalues.

    $$A=\left[\begin{array}{cc}6& 0\\ 2& -1\end{array}\right]$$

    Answer: $\left\{-1,6\right\}$

    Step-by-step solution

    1. $$\det (\lambda I-A)={\lambda }^2-5\lambda -6$$
    2. $$\lambda \in \left\{-1,6\right\}$$
16. 16.

    Find the distinct eigenvalues.

    $$A=\left[\begin{array}{cc}3& 4\\ 1& 3\end{array}\right]$$

    Answer: $\left\{1,5\right\}$

    Step-by-step solution

    1. $$\det (\lambda I-A)={\lambda }^2-6\lambda +5$$
    2. $$\lambda \in \left\{1,5\right\}$$
17. 17.

    Find the distinct eigenvalues.

    $$A=\left[\begin{array}{cc}2& 0\\ 7& 5\end{array}\right]$$

    Answer: $\left\{2,5\right\}$

    Step-by-step solution

    1. $$\det (\lambda I-A)={\lambda }^2-7\lambda +10$$
    2. $$\lambda \in \left\{2,5\right\}$$
18. 18.

    Find the distinct eigenvalues.

    $$A=\left[\begin{array}{cc}8& 0\\ 0& 1\end{array}\right]$$

    Answer: $\left\{1,8\right\}$

    Step-by-step solution

    1. $$\det (\lambda I-A)={\lambda }^2-9\lambda +8$$
    2. $$\lambda \in \left\{1,8\right\}$$

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• Orthogonal Projections
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