Linear Algebra with Applications, 10th Edition

Section 6.1 Exercises

1. Find the eigenvalues and the corresponding eigenspaces for each of the following matrices:

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

   2. $\left[\begin{array}{cc}6& -4\\ 3& -1\end{array}\right]$

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

   4. $\left[\begin{array}{cc}3& \hfill -8\\ 2& \hfill 3\end{array}\right]$

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

   6. $\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]$

   7. $\left[\begin{array}{ccc}1& 1& 1\\ 0& 2& 1\\ 0& 0& 1\end{array}\right]$

   8. $\left[\begin{array}{ccc}1& 2& \hfill 1\\ 0& 3& \hfill 1\\ 0& 5& \hfill -1\end{array}\right]$

   9. $\left[\begin{array}{rrr}4& -5& \hfill 1\\ 1& 0& \hfill -1\\ 0& 1& \hfill -1\end{array}\right]$

   10. $\left[\begin{array}{rrr}-2& 0& \hfill 1\\ 1& 0& \hfill -1\\ 0& 1& \hfill -1\end{array}\right]$

   11. $\left[\begin{array}{cccc}2& 0& 0& 0\\ 0& 2& 0& 0\\ 0& 0& 3& 0\\ 0& 0& 0& 4\end{array}\right]$

   12. $\left[\begin{array}{cccc}3& 0& 0& 0\\ 4& 1& 0& 0\\ 0& 0& 2& 1\\ 0& 0& 0& 2\end{array}\right]$

2. Show that the eigenvalues of a triangular matrix are the diagonal elements of the matrix.

3. Let A be an $n\times n$ matrix. Prove that A is singular if and only if $\lambda =0$ is an eigenvalue of A.

4. Let A be a nonsingular matrix and let $\lambda$ be an eigenvalue of A. Show that $1/\lambda$ is an eigenvalue of $A^{-1}$.

5. Let A and B be $n\times n$ matrices. Show that if none of the eigenvalues of A are equal to 1, then the matrix equation

   $$XA+B=X$$

   will have a unique solution.

6. Let $\lambda$ be an eigenvalue of A and let x be an eigenvector ...