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\textbf{Assignment 18:  Matrix equations and matrix algebra}

\begin{enumerate}

\item For the two matrices below, find the dimension of the solution
  space of $A \bx = \bz$. Then find all the solutions.  You should be
  able to determine the dimension without finding the solutions.

$A = \begin{pmatrix}1&2&0\\0&0&1\\0&0&0\end{pmatrix}$ and 
$A = \begin{pmatrix}1&2&0&0&3\\0&0&1&0&0\\0&0&0&1&2\end{pmatrix}$

\item Suppose that the matrix $A$ has row-reduced echelon form $R$ given
  below.

$A =
\begin{pmatrix}1&2&3&4&5\\6&7&8&9&1\\2&3&4&5&6\\7&8&9&1&2\end{pmatrix}$
and
$R =
\begin{pmatrix}1&0&-1&0&0\\0&1&2&0&0\\0&0&0&1&0\\0&0&0&0&1\end{pmatrix}$

Find all solutions to the matrix equation 
$A\bx = \begin{pmatrix}6\\7\\8\\9\end{pmatrix}$ given that 
$\bx = \begin{pmatrix}1\\0\\0\\0\\1\end{pmatrix}$ is a particular
solution.

\item Let $A = \begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}$,
$B = \begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}$, and
$C = \begin{pmatrix}1&2\\3&4\\5&6\end{pmatrix}$.

Determine which of the nine products $A^2, AB, AC, BA, B^2, BC, CA,
CB, C^2$ are defined.  Evaluate the first three valid products.

\item Find $\begin{pmatrix}1&1\\0&1\end{pmatrix}^{1234}$.  You may
  want to find  $\begin{pmatrix}1&1\\0&1\end{pmatrix}^{2}$ and 
 $\begin{pmatrix}1&1\\0&1\end{pmatrix}^{3}$ to start.
 Can you prove your result?

\item Find two $2 \times 2$ matrices $A$ and $B$ neither of which is
  the zero matrix, but for which $AB = \bz$.

\end{enumerate}


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