<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="description">
    <name>
      <text>7.3.30 (Notes on Gaussian elimination)</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<h3>Some notes</h3>

<p>The method of Gaussian elimination actually an algorithmic method to find the inverse of a matrix. This allows us to solve a wider range of systems of equations. When solving a system of equations we stop performing row operations when the augmented matrix of the system has the following properties.</p>
<ol>
<li>The first non-zero element in each row, called the <b>leading entry</b>, is \(1\).</li>
<li>Each leading entry is in a column to the right of the leading entry in the previous row.</li>
<li>Rows with all zero elements, if any, are below rows having a non-zero element.</li>
<li>The leading entry in each row is the only non-zero entry in its column.</li>
</ol>
<p>If a matrix satisfies these properties, then the matrix is said to be in <b> reduced row echelon form</b>. The process of Gaussian elimination is in general the fastest way to solve larger systems of equations as it is easily generalised to matrices of larger sizes.</p>
<p>In summary, this means that in order to solve a system of linear equations we do the following:</p>
<ol>
<li>Put the system of linear equations in the form \(A \mathbf{x} = \mathbf{b}\) where \(A\) is the matrix of coefficients, \(\mathbf{x}\) is the column vector of variables, and \(\mathbf{b}\) is the column vector of the values of the rhs of each equals sign.</li>
<li>Form the augmented matrix of the system by appending the vector \(\mathbf{b}\) to the right hand side of the matrix \(A\).</li>
<li>Perform Gaussian elimination on the augmented matrix until it is in reduced row echelon form.</li>
<li>Read off the solutions of the system of equations.</li>
</ol>]]></text>
    </questiontext>
    <generalfeedback format="moodle_auto_format">
      <text/>
    </generalfeedback>
    <defaultgrade>0</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
  </question>
</quiz>
