1.
\(\A=\left(\begin{array}{rr}2\amp 0\\0\amp 3 \end{array} \right)\)
Section 8.3 Eigenvalues and Eigenvectors: Homework
Exercises Exercises
Exercise Group.
Solve the eigenproblem for each of the matrices.
2.
\(\A=\left(\begin{array}{rr}2\amp 0\\0\amp -5 \end{array} \right)\)
3.
\(\A=\left(\begin{array}{rr}3\amp 0\\0\amp 3 \end{array} \right)\)
4.
\(\A=\left(\begin{array}{rr}0\amp 1\\0\amp 0 \end{array} \right)\)
5.
\(\A=\left(\begin{array}{rr}9\amp -30\\4\amp -13 \end{array} \right)\)
6.
\(\A=\left(\begin{array}{rr}14\amp -45\\2\amp -5 \end{array} \right)\)
7.
\(\A=\left(\begin{array}{rr}7\amp -4\\12\amp -7 \end{array} \right)\)
8.
\(\A=\left(\begin{array}{rr}11\amp 10\\-3\amp 0 \end{array} \right)\)
9.
\(\A=\left(\begin{array}{rr}0\amp -9\\-4\amp 0 \end{array} \right)\)
10.
\(\A=\left(\begin{array}{rrr}-7\amp 6\amp 0\\-9\amp 8\amp 0\\18\amp -12\amp 2 \end{array} \right)\)
11.
\(\A=\left(\begin{array}{rrr}4\amp 5\amp -17\\9\amp 16\amp -51\\3\amp 5\amp -16 \end{array} \right)\)
12.
Find a matrix \(\A\) whose eigenvalues are \(-2,3\) with corresponding eigenvectors \(\left(\begin{array}{r}5\\-4 \end{array} \right),\left(\begin{array}{r}-4\\3 \end{array} \right)\text{.}\)
13.
Find a matrix \(\A\in \R^{2 \times 2}\) whose eigenvalues are \(3,-4\) with corresponding eigenvectors \(\left(\begin{array}{r}-3\\2 \end{array} \right),\left(\begin{array}{r}4\\-3 \end{array} \right)\text{.}\)
14.
Suppose \(\lambda\) is an eigenvalue, and \(\x\) a corresponding eigenvector, of \(\A\text{.}\) Prove that \(\lambda^2\) is an eigenvalue, and \(\x\) a corresponding eigenvector, of \(\A^2\text{.}\)
15.
Suppose \(\lambda\) is an eigenvalue, and \(\x\) a corresponding eigenvector, of the nonsingular matrix \(\A\text{.}\) Prove that \(\displaystyle{\frac{1}{\lambda}}\) is an eigenvalue, and \(\x\) a corresponding eigenvector, of \(\A^{-1}\text{.}\)
16.
For each matrix \(\A\) from Exercise Group 8.3.1–11, determine if the matrix is diagonalizable. If so write it as \(\A=\X\Lam\X^{-1}\text{.}\)
Exercise Group.
Show all work and computations for this problem (i.e. do by hand yourself). For each matrix \(\A\text{:}\)
- Show \(\A\) is diagonalizable by writing it as \(\A=\X\Lam\X^{-1}\text{.}\)
- Use the eigenvalues to find the determinant of \(\A\text{.}\)
17.
\(\A=\left(\begin{array}{rr}3 \amp 2 \\-2 \amp -2 \end{array} \right)\)
18.
\(\A=\begin{pmatrix}2 \amp 1\\1 \amp 2 \end{pmatrix}\)
19.
\(\A=\left(\begin{array}{rrr}1\amp 1\amp 1\\ 0\amp 2\amp 1\\ 0\amp 0\amp 3 \end{array} \right)\)
20.
\(\A=\left(\begin{array}{rrr}2\amp 2\amp 0\\ 1\amp 1\amp 2\\ 1\amp 1\amp 2 \end{array} \right)\)
21.
Suppose \(\A\) is diagonalizable with \(\A=\X\Lam\X^{-1}.\) Write \(\A^2\) in terms of \(\X,\X^{-1}\text{,}\) and \(\Lam.\) Do the same for \(\A^3\text{,}\) \(\A^4\text{,}\) and \(\A^n\text{.}\)
22.
Let \(\A=\left(\begin{array}{cc} 0.2 \amp 0.3 \\0.8 \amp 0.7 \end{array} \right).\) Such a matrix is called a (left) stochastic matrix because the sum of the entries in each column is \(1.\) Show that \(1\) is an eigenvalue for \(\A\text{.}\)
23.
Let \(\A=\left(\begin{array}{cc} a\amp b\\c\amp d \end{array} \right)\) be a (left) stochastic matrix so that \(a+c=1\) and \(b+d=1\text{.}\) Show that \(1\) is an eigenvalue for \(\A\text{.}\)
24.
Use the ideas in this section to easily find a matrix whose eigenvalues are \(3,-4\) with corresponding eigenvectors \(\left(\begin{array}{r}-3\\2 \end{array} \right), \left(\begin{array}{r}4\\-3 \end{array} \right)\text{,}\) repsectively. Compare your answer to that of Problem 8.3.13.
25.
Recall your \(\A=\X\Lam X^{-1}\) from Problem 8.3.24. Reverse the order of the eigenvalues in the matrix \(\Lam\) and the order of the corresponding eigenvectors in \(\X\) to get a new \(\widetilde{\X},\widetilde{\Lam}\text{,}\) and \(\widetilde{\X}^{-1}\text{.}\) Does \(\widetilde{\X}\widetilde{\Lam}\widetilde{\X}^{-1}\) equal the \(\A\) you found in Problem 8.3.24?
26.
Use the ideas in this section to easily find a matrix whose eigenvalues are \(1, 2\text{,}\) and \(-5\) with corresponding eigenvectors \(\left(\begin{array}{r}1\\2\\0 \end{array} \right)\text{,}\) \(\left(\begin{array}{r}-3\\-5\\-2 \end{array} \right)\text{,}\) and \(\left(\begin{array}{r}11\\19\\7 \end{array} \right)\text{,}\) respectively. You may use an app such as Wolfram Alpha or a website such as
https://matrix.reshish.com to compute inverses or do matrix multiplication. Then check your result using Wolfram Alpha or a website such as https://www.symbolab.com/solver/matrix-eigenvalues-calculator. If using a website just be sure to write down the answers to all the calculation you do and a give a citation of the website you use.
