Calculating Determinants

Calculating Determinants

The determinant of a $2 \times 2$ matrix

$\begin{displaymath} M = \left( \begin{array}{cccc} m_{11} & m_{12} \\ m_{21} & m_{22} \end{array}\right) \end{displaymath}$ (1)

is given by

$\begin{displaymath} \vert M\vert = \left\vert \begin{array}{cccc} m_{11} & m_{1... ... \end{array}\right\vert = m_{11} \, m_{22} - m_{12} \, m_{21} \end{displaymath}$ (2)
The determinant of a $3 \times 3$ matrix

$\begin{displaymath} M = \left( \begin{array}{cccc} m_{11} & m_{12} & m_{13} \\ ... ...{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{array}\right) \end{displaymath}$ (3)

can be written in terms of the determinants of $2 \times 2$ sub-matrices

$\displaystyle \vert M\vert$ $\textstyle =$ $\displaystyle \left\vert \begin{array}{cccc} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{array}\right\vert$ (4)

$\textstyle =$ $\displaystyle m_{11} \left\vert \begin{array}{cccc} m_{22} & m_{23} \\ m_{32} ... ... \begin{array}{cccc} m_{21} & m_{22} \\ m_{31} & m_{32} \end{array}\right\vert$
In general, each element of the top row of the matrix is multiplied by the determinant of the sub-matrix obtained by removing the row and column containing that element. The results are then added together with alternating sign, starting with a positive $m_{11}$ term. Figure 1 shows the top-row elements and the associated sub-matrices and signs for the $2 \times 2$ , $3 \times 3$ , and $4 \times 4$ cases.

Figure 1: A visual guide to computing the determinants of $2 \times 2$ , $3 \times 3$ , and $4 \times 4$ matrices.

$\scalebox{0.6}{ \includegraphics{dets.eps} }$

**Figure 1:** A visual guide to computing the determinants of $2 \times 2$ , $3 \times 3$ , and $4 \times 4$ matrices.
$\scalebox{0.6}{ \includegraphics{dets.eps} }$

An $\mathbf{n=2}$ Example

$\displaystyle M$	$\textstyle =$	$\displaystyle \left( \begin{array}{cccc} 3 & -1 \\ 5 & 7 \end{array}\right)$
$\displaystyle \vert M\vert$	$\textstyle =$	$\displaystyle 3 \cdot 7 - (-1)\cdot 5 = 21 + 5$	(5)
	$\textstyle =$	$\displaystyle 26$	(6)

An $\mathbf{n=3}$ Example

$\displaystyle M$	$\textstyle =$	$\displaystyle \left( \begin{array}{cccc} 4 & 2 & 5 \\ -1 & 6 & 7 \\ 3 & 1 & 2 \end{array}\right)$
$\displaystyle \vert M\vert$	$\textstyle =$	$\displaystyle 4 \left\vert \begin{array}{cccc} 6 & 7 \\ 1 & 2 \end{array}\righ... ... \,+\, 5 \left\vert \begin{array}{cccc} -1 & 6 \\ 3 & 1 \end{array}\right\vert$
	$\textstyle =$	$\displaystyle 4 \, (6 \cdot 2 - 7 \cdot 1) - 2 \, ((-1) \cdot 2 - 7 \cdot 3) + 5 \, ((-1) \cdot 1 - 6 \cdot 3)$
	$\textstyle =$	$\displaystyle 20 + 46 - 95$
	$\textstyle =$	$\displaystyle -29$	(7)

Updated Mon Jan 19 13:29:10 2004

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$\displaystyle \vert M\vert$	$\textstyle =$	$\displaystyle \left\vert \begin{array}{cccc} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{array}\right\vert$	(4)
	$\textstyle =$	$\displaystyle m_{11} \left\vert \begin{array}{cccc} m_{22} & m_{23} \\ m_{32} ... ... \begin{array}{cccc} m_{21} & m_{22} \\ m_{31} & m_{32} \end{array}\right\vert$

An Example

An Example

An $\mathbf{n=2}$ Example

An $\mathbf{n=3}$ Example