# 4.2.1: Exercises 4.2

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In Exercises $$\PageIndex{1}$$ – $$\PageIndex{6}$$, a matrix $$A$$ is given. For each,

1. Find the eigenvalues of $$A$$, and for each eigenvalue, find an eigenvector.
2. Do the same for $$A^{T}$$.
3. Do the same for $$A^{−1}$$
4. Find $$\text{tr}(A)$$.
5. Find $$\text{det}(A)$$. Use Theorem 4.2.1 to verify your results.
##### Exercise $$\PageIndex{1}$$

$$\left[\begin{array}{cc}{0}&{4}\\{-1}&{5}\end{array}\right]$$

1. $$\lambda_{1}=1\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{4}\\{1}\end{array}\right];$$
$$\lambda_{2}=4\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{1}\\{1}\end{array}\right]$$
2. $$\lambda_{1}=1\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-1}\\{1}\end{array}\right];$$
$$\lambda_{2}=4\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{-1}\\{4}\end{array}\right]$$
3. $$\lambda_{1}=1/4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{1}\\{1}\end{array}\right];$$
$$\lambda_{2}=1\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{4}\\{1}\end{array}\right]$$
4. $$5$$
5. $$4$$
##### Exercise $$\PageIndex{2}$$

$$\left[\begin{array}{cc}{-2}&{-14}\\{-1}&{3}\end{array}\right]$$

1. $$\lambda_{1}=-4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{7}\\{1}\end{array}\right];$$
$$\lambda_{2}=5\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{-2}\\{1}\end{array}\right]$$
2. $$\lambda_{1}=-4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{1}\\{2}\end{array}\right];$$
$$\lambda_{2}=5\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{-1}\\{7}\end{array}\right]$$
3. $$\lambda_{1}=-1/4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{7}\\{1}\end{array}\right];$$
$$\lambda_{2}=1/5\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{-2}\\{1}\end{array}\right]$$
4. $$1$$
5. $$-20$$
##### Exercise $$\PageIndex{3}$$

$$\left[\begin{array}{cc}{5}&{30}\\{-1}&{-6}\end{array}\right]$$

1. $$\lambda_{1}=-1\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-5}\\{1}\end{array}\right];$$
$$\lambda_{2}=0\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{-6}\\{1}\end{array}\right]$$
2. $$\lambda_{1}=-1\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{1}\\{6}\end{array}\right];$$
$$\lambda_{2}=0\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{1}\\{5}\end{array}\right]$$
3. $$A$$ is not invertible.
4. $$-1$$
5. $$0$$
##### Exercise $$\PageIndex{4}$$

$$\left[\begin{array}{cc}{-4}&{72}\\{-1}&{13}\end{array}\right]$$

1. $$\lambda_{1}=4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{9}\\{1}\end{array}\right];$$
$$\lambda_{2}=5\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{8}\\{1}\end{array}\right]$$
2. $$\lambda_{1}=4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-1}\\{8}\end{array}\right];$$
$$\lambda_{2}=5\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{-1}\\{9}\end{array}\right]$$
3. $$\lambda_{1}=1/4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{9}\\{1}\end{array}\right];$$
$$\lambda_{2}=1/5\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{8}\\{1}\end{array}\right]$$
4. $$9$$
5. $$20$$
##### Exercise $$\PageIndex{5}$$

$$\left[\begin{array}{ccc}{5}&{-9}&{0}\\{1}&{-5}&{0}\\{2}&{4}&{3}\end{array}\right]$$

1. $$\lambda_{1}=-4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-7}\\{-7}\\{6}\end{array}\right];$$
$$\lambda_{2}=3\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right]$$
$$\lambda_{3}=4\text{ with }\vec{x_{3}}=\left[\begin{array}{c}{9}\\{1}\\{22}\end{array}\right]$$
2. $$\lambda_{1}=-4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-1}\\{9}\\{0}\end{array}\right];$$
$$\lambda_{2}=3\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{-20}\\{26}\\{7}\end{array}\right]$$
$$\lambda_{3}=4\text{ with }\vec{x_{3}}=\left[\begin{array}{c}{-1}\\{1}\\{0}\end{array}\right]$$
3. $$\lambda_{1}=-1/4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-7}\\{-7}\\{6}\end{array}\right];$$
$$\lambda_{2}=1/3\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right]$$
$$\lambda_{3}=1/4\text{ with }\vec{x_{3}}=\left[\begin{array}{c}{9}\\{1}\\{22}\end{array}\right]$$
4. $$3$$
5. $$-48$$
##### Exercise $$\PageIndex{6}$$

$$\left[\begin{array}{ccc}{0}&{25}&{0}\\{1}&{0}&{0}\\{1}&{1}&{-3}\end{array}\right]$$

1. $$\lambda_{1}=-5\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-5}\\{1}\\{2}\end{array}\right];$$
$$\lambda_{2}=-3\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right]$$
$$\lambda_{3}=5\text{ with }\vec{x_{3}}=\left[\begin{array}{c}{20}\\{4}\\{3}\end{array}\right]$$
2. $$\lambda_{1}=-5\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-1}\\{5}\\{0}\end{array}\right];$$
$$\lambda_{2}=-3\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{1}\\{-11}\\{8}\end{array}\right]$$
$$\lambda_{3}=5\text{ with }\vec{x_{3}}=\left[\begin{array}{c}{1}\\{5}\\{0}\end{array}\right]$$
3. $$\lambda_{1}=-1/5\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-5}\\{1}\\{2}\end{array}\right];$$
$$\lambda_{2}=-1/3\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right]$$
$$\lambda_{3}=1/5\text{ with }\vec{x_{3}}=\left[\begin{array}{c}{20}\\{4}\\{3}\end{array}\right]$$
4. $$-3$$
5. $$75$$

4.2.1: Exercises 4.2 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.