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\textbf{Assignment 19: More matrix equations and inverses}

\begin{enumerate}
\item Show that
  $\begin{pmatrix}1&2&3&4&5\\6&7&8&9&1\\2&3&4&5&6\\7&8&9&1&2\end{pmatrix}
  \begin{pmatrix}
    x_1\\x_2\\x_3\\x_4\\x_5
  \end{pmatrix} =
  \begin{pmatrix}
    b_1\\b_2\\b_3\\b_4
  \end{pmatrix}$ is solvable for all choice of $b_i$.  Hint: the
  row-reduced echelon form of the $4\times 5 $ matrix is $R =
\begin{pmatrix}1&0&-1&0&0\\0&1&2&0&0\\0&0&0&1&0\\0&0&0&0&1\end{pmatrix}$.

Moreover, show that for each choice of $\begin{pmatrix}
  b_1\\b_2\\b_3\\b_4
  \end{pmatrix}$, there are infinitely many solutions.
\item Find all $\begin{pmatrix} b_1\\b_2\\b_3\\b_4\\b_5
  \end{pmatrix}$ for which 

  $\begin{pmatrix}1&0&0&0\\2&0&0&0\\0&3&0&0\\0&4&0&0\\0&0&5&0\end{pmatrix}
\begin{pmatrix}
  x_1\\x_2\\x_3\\x_4
  \end{pmatrix} =
  \begin{pmatrix}
    b_1\\b_2\\b_3\\b_4\\b_5
  \end{pmatrix}$ is solvable.
  
  Find all solutions to $
  \begin{pmatrix}1&0&0&0\\2&0&0&0\\0&3&0&0\\0&4&0&0\\0&0&5&0\end{pmatrix}
\begin{pmatrix}
  x_1\\x_2\\x_3\\x_4
  \end{pmatrix} =\begin{pmatrix}
    1\\2\\3\\4\\5
  \end{pmatrix}$ 
  
\item Let $A = \begin{pmatrix}1&0&1\\0&1&2\\1&2&4\end{pmatrix}$.
  
  Use row reduction to find the inverse of $A$, and solve the matrix
  equation $A \bx = \begin{pmatrix} 1\\2\\3
  \end{pmatrix}$
  
  both by using the inverse of $A$, and by row reducing the augmented
  matrix.

\end{enumerate}


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