<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[11pt]{amsart}
%Specify document type amsart, 11pt font

\usepackage{amsmath, amssymb}
%packages for special characters and typesetting

\usepackage{mathtools}
%tools for typesetting matrices such as bmatrix*


\theoremstyle{definition}
%specify style of theorem environment

\newtheorem{problem}{Problem}
%creates theorem environment "Problem"

\newtheorem{solution}{Solution}





\begin{document}

\title{MATH 351, Fall 2017, Homework \#5}

\author{\underline{\bf Due Friday, October 6, 2017}}



\maketitle

\begin{problem}[cf. 3.64, p. 190] Invert the following matrices, if possible.
\end{problem}

\begin{itemize}
\item[a)] $$\begin{bmatrix*}[r] 0 &amp; 1 &amp; 1\\ 1 &amp; 0 &amp; 1\\ 1 &amp; 1 &amp; 0\end{bmatrix*}$$

\item[b)] $$\begin{bmatrix*}[r] 2 &amp; -1 &amp; 2 &amp; 0\\ 4 &amp; -1 &amp; 4 &amp; -2\\ 8 &amp; -3 &amp; 10 &amp; 0\\ 6 &amp; -3 &amp; 8 &amp; 9\end{bmatrix*}$$

\item[c)] $$\begin{bmatrix*}[r] 5 &amp; -1 &amp; 2\\ 1 &amp; 0 &amp; 1\\ 6 &amp; 0 &amp; 3\end{bmatrix*}$$

\end{itemize}




\begin{problem}[cf. 3.74, p. 192]. A matrix $N\in M_{nn}(\mathbb R)$ is said to be {\bf nilpotent} of degree $k$ if $N^k = 0$. 
\end{problem}

\begin{itemize}
\item[a)] If $N$ is nilpotent of degree $k$, show that $$(I - N)^{-1} = I + N + N^2 + \dotsb + N^{k-1}.$$

\item[b)] Use this to invert the matrix $$\begin{bmatrix*}[r] 1 &amp; -2 &amp; 3\\ 0 &amp; 1 &amp; 4\\ 0 &amp; 0 &amp; 1\end{bmatrix*}$$

\end{itemize}


\begin{problem}[3.77, p. 192] If $A$ is a square matrix such that $A^2 + 3A + I = 0$, show that $-A - 3I$ is an inverse for $A$.
\end{problem}



\begin{problem}[3.80 and 3.84, p. 193] If $A, B\in M_{nn}(\mathbb R)$ are invertible matrices.
\end{problem}

\begin{itemize}
\item[a)] Show that $AB$ is again invertible.

\item[b)] Show that $(A^{-1})^t = (A^t)^{-1}$.
\end{itemize}




\begin{problem}[cf. 2.3, p. 109] Find a basis for ${\rm Col}(A)$.
\end{problem}

\begin{itemize}
\item[a)] $$\begin{bmatrix*}[r] 1 &amp; 2 &amp; -8 &amp; 4\\ 2 &amp; 1 &amp; -1 &amp; -1\\ -3 &amp; 3 &amp; -21 &amp; 15\\ 3 &amp; 4 &amp; -14 &amp; 6\end{bmatrix*}$$

\item[b)] $$\begin{bmatrix*}[r] 2 &amp; 3 &amp; -1 &amp; 19 &amp; 2\\ -1 &amp; 0 &amp; 2 &amp; -8 &amp; 5\\ 2 &amp; -1 &amp; -5 &amp; 15 &amp; -14\end{bmatrix*}$$

\item[c)] $$\begin{bmatrix*}[r] 3 &amp; 1 &amp; 2 &amp; -7\\ 2 &amp; 4 &amp; 0 &amp; 8\\ -1 &amp; -2 &amp; -1 &amp; -2\\ 2 &amp; 2 &amp; 3 &amp; 8\end{bmatrix*}$$

\end{itemize}

\begin{problem}[cf. 2.10, p. 111] Let $A\in M_{kn}(\mathbb R)$. Is there relationship between the rank of the homogeneous system of linear equations given by $A\cdot\vec x = \vec 0$ and the number of vectors in a basis for ${\rm Col}(A)?$. Explain.
\end{problem}


\begin{problem}[2.20, p. 112] Show that the columns of a matrix are linearly independent exactly when the null space of $A$ is $\{\vec 0\}$.
\end{problem}

\begin{problem}[cf. 2.41, p. 125] Consider $\mathbb R^n$
\end{problem}

\begin{itemize}
\item[a)] Find a basis for $\mathbb R^4$ containing the vectors $(1,2,3,4)^t$ and $(1,0, 1, 0)^t$.

\item[b)] Given $k$ linearly independent vectors in $\mathbb R^n$, is it always possible to find a basis for $\mathbb R^n$ containing those vectors? Explain.

\end{itemize}



\end{document}



</pre></body></html>