Question 5. Let A ? R<sup>n×n</sup> be a stochastic matrix. Use the fact that A is stochastic, i.e., all columns sum to one,
to demonstrate that A must have an eigenvalue ? = 1.
Question 6. Let
$\begin{bmatrix} r & s & t \\ 0 & u & v \\ 0 & 0 & w \end{bmatrix}$ ? R<sup>3×3</sup> b = $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$ ? R<sup>3</sup>.
We know that the system b = Ax, for x ? R<sup>3</sup>, has (a) a unique solution, (b) infinitely many solutions, or (c) no
solutions. Give conditions on r, s,..., z ? R so that the system b = Ax has
(a) a unique solution,
(b) infinitely many solutions, or
(c) no solutions.