     \end{vmatrix}= ad-bc.
    \end{align*}
    
    The determinant of a $3\times 3$ matrix is the number

    <span class="marginpar">
      Notice the negative sign on the middle term of the $3 \times
      3$ determinant. Also, notice that we had to compute three
      determinants of 2 by 2 matrices in order to find the
      determinant of a 3 by 3 matrix.
    </span>

\begin{align*}
\begin{vmatrix}a&amp;b&amp;c\\d&amp;e&amp;f\\g&amp;h&amp;i\end{vmatrix} &amp;= a\det\begin{vmatrix}e&amp;f\\h&amp;i\end