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\textbf{Assignment on Systems of linear Equations}

\begin{enumerate}
\item Write each system of linear equations in matrix form:

  \begin{alignat*}{2}
    x + y &= 1\hspace{1in}3x + y + z &= 1\\
    y - z &= 1\hspace{1in}x - y - z &= 0\\
    x + z &= 0
  \end{alignat*}
  
\item Write a system of linear equations equivalent to the matrix
  equation

$  \begin{pmatrix}
    1&2&3&4\\5&6&7&8\\9&10&11&12
  \end{pmatrix}
  \begin{pmatrix}
    w\\x\\y\\z
  \end{pmatrix}
  =
  \begin{pmatrix}
    13\\14\\15
  \end{pmatrix}
$

\item Solve the systems in the first problem by a sequence of
  elementary operations applied to both sides of the equations.

\item Row reduce the following matrices to echelon form and solve the
  associated matrix equations: $A \mathbf{x} = \mathbf{0}$. 

$
  \begin{pmatrix}
    1&2&0\\0&0&0\\0&0&1
  \end{pmatrix}
  \begin{pmatrix}
    1&0&0\\0&2&0\\0&1&1
  \end{pmatrix}
  \begin{pmatrix}
    1&2&0&0\\
    0&0&1&3\\
    0&0&3&9
  \end{pmatrix}
  \begin{pmatrix}
    1&2&3&4\\
    0&5&6&7\\
    0&0&8&9\\
    0&0&0&10
  \end{pmatrix}
$
\end{enumerate}

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