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\pagestyle{fancy} \lhead{{\sf Homework 3 $|$ Due January 31 (Friday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 311 $|$ Linear Algebra $|$ Spring 2014 
    $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Strang, Section 2.4, 2.5, 2.6, 2.7 \\

\noindent\emph{Suggested short conceptual exercises}: Strang, Section
2.4, \#2, 7, 11, 14, 32, 33. Section 2.5, \#11, 15, 29. Section 2.7,
\#3--5, 8, 11, 12--16, 19. \\


% Source: Unless otherwise noted, wrote all problems from scratch.

\begin{enumerate}


\item Let $\A$, $\B$, and $\C$ be $n\times n$ matrices.
  \begin{enumerate}
  \item If $\A$ is invertible and $\A\B=\A\C$, prove that $\B=\C$.
  \item Find an example of three nonzero matrices such that
    $\A\B=\A\C$ but $\B\neq\C$.
  \item Suppose that $\B=\C^{-1}\A\C$ is invertible. Find formulas for
    $\A$, $\A^{-1}$, and $\B^{-1}$.
  \end{enumerate}  



% Strang: Section 2.5, #7 (modified)
\item Let $\A$ be a $3\times 3$ matrix where row 1 + row 2 = row
  3. Give elementary answers to the following questions (that is, do
  not use terms such as ``column space'' or ``nullspace''.)
\begin{enumerate}
  \item Explain why $\A\x=\vvv{1}{0}{0}$ cannot have a solution. 
  \item Which vectors $\vvv{b_1}{b_2}{b_3}$ might allow a solution to
    $\A\x=\b$? [\emph{Hint}: Find a linear equation relating $b_1$,
    $b_2$, and $b_3$.]
  \item What happens to row $3$ in elimination?
\end{enumerate}



% Strang: Section 2.5, #8 (modified)
\item Let $\A$ be a $3\times 3$ matrix where column 1 + column 2 =
  column 3.
  \begin{enumerate}
  \item Find a nonzero solution to $\A\x=\0$.
  \item Elimination keeps column 1 + column 2 = column 3. Explain why
    there is no third pivot. Conclude that $\A$ is not invertible.
  \end{enumerate}
  
    
  
\item Let $\A$ be an invertible $3\times 3$ matrix, and let $\B$ be
  the matrix obtained from $\A$ by taking the bottom row and making it
  the top row instead (and so the first and second rows of $\A$ become
  the second and third rows of $\B$, respectively).
  \begin{enumerate}
  \item Find the permutation matrix $\P$ that you need to multiply $\A$ by
    to get $\B$. Do you need to multiply on the left or on the right?
  \item What is $\P^{-1}$?
  \item Find simple formulas for $\B^{-1}$ and $\B^T$.
  \end{enumerate}

  

% Strang: Section 2.6, #7 (modified)
\item Consider the matrix $\A=\begin{bmatrix} 1 & 0 & 1 \\ 2 & 2 & 2
  \\ 3 & 4 & 5\end{bmatrix}$.
  \begin{enumerate}
  \item What three elementary matrices $\E_{21}$, $\E_{31}$, and
    $\E_{32}$ put $\A$ into its upper triangular form,
    $\E_{32}\E_{31}\E_{21}\A=\U$?
  \item Multiply by $\E_{32}^{-1}$, $\E_{31}^{-1}$ and $\E_{21}^{-1}$
    to factor $\A$ into $\A=\L\U$, where
    $\L=\E_{21}^{-1}\E_{31}^{-1}\E_{32}^{-1}$ is a lower triangular
    matrix whose entries are the \emph{multipliers} of elimination and
    $\U$ is an upper triangular matrix with the pivots on the
    diagonal.
  \item Factor $\U=\D\U_1$, where $\D$ is a diagonal matrix that
    contains the pivots.
  \item Factor $\A=\L\D\U_1$, where $\L$ is a lower triangular matrix
    with $1$s on the diagonal and whose other entries are the
    multipliers, $\D$ is a diagonal matrix that contains the pivots,
    and $\U_1$ is an upper triangular matrix with $1$s on the
    diagonal.
  \end{enumerate}



\item Let $\A$ be a $3\times 3$ matrix. Suppose you want to do the
  following operations on $\A$: Subtract row 1 from row 2, subtract
  row 1 from row 3, and subtract row 2 from row 3.
  \begin{enumerate}
  \item First, suppose you want to perform these operations on $\A$
    \emph{simultaneously}. Write down a matrix $\B$ that you need to
    multiply $\A$ by to achieve this. Does $\A\B$ or $\B\A$ yield the
    desired result? What is $\B^{-1}$? [\emph{Hint}: You should be
      able to find $\B^{-1}$ by inspection, but it's not quite a
      simple as flipping the signs of the entries below the diagonal.]
  \item Now, suppose you want to perform these operations on $\A$
    \emph{sequentially}. Write down a matrix $\E$ that you need to
    multiply $\A$ by to achieve this. It should be the product of
    three elementary matrices [\emph{Hint}: It should \emph{not} the
      same matrix as $B$ from Part~(a)]. What is $\E^{-1}$.
  \item Suppose you factor $\A=\L\U$, where $\U$ is the upper
    triangular matrix with the pivots on the diagonal, and $\L$ is
    lower-triangular. What is the relationship between $\L$ and $\E$?
  \end{enumerate}



% Strang: Section 2.6: #20 (modified)
\item A matrix is \emph{symmetric} if $\A^T=\A$. Factor the following
  matrices into $\A=\L\D\L^T$.
  \[
  \A=\begin{bmatrix}1 & 3 \\ 3 & 2\end{bmatrix}\,,\qquad\qquad
  \A=\begin{bmatrix}1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2\end{bmatrix} 
  \]
  Recall that for any symmetric matrix, the $\A=\L\D\U$ factorization
  simplifies to $\A=\L\D\L^T$.


\end{enumerate}

\end{document}
