Linear Algebra with Applications, 10th Edition

Section 6.8 Exercises

1. Find the eigenvalues of each of the following matrices and verify that conditions (i), (ii), and (iii) of Theorem 6.8.1 hold:

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

   2. $\left[\begin{array}{cc}4& 2\\ 2& 7\end{array}\right]$

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

2. Find the eigenvalues of each of the following matrices and verify that conditions (i) and (ii) of Theorem 6.8.2 hold:

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

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

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

3. Find the output vector x in the open version of the Leontief input–output model if

   $$\begin{array}{ccc}A=\left[\begin{array}{ccc}0.2& 0.4& 0.4\\ 0.4& 0.2& 0.2\\ 0.0& 0.2& 0.2\end{array}\right]& \text{and}& \mathbf{d}=\left[\begin{array}{c}\hfill 16,000\\ \hfill 8,000\\ \hfill 24,000\end{array}\right]\end{array}$$

4. Consider the closed version of the Leontief input–output model with input matrix

   $$A=\left[\begin{array}{ccc}0.5& 0.4& 0.1\\ 0.5& 0.0& 0.5\\ 0.0& 0.6& 0.4\end{array}\right]$$

   If $\mathbf{x}={(x_1,x_2,x_3)}^T$ is any output vector for this model, how are the coordinates $x_1,x_2$, and $x_3$ related?

5. Prove: If $A^m=O$ for some positive integer m, then $I-A$ is nonsingular. ...