Matrix Determinant

Given a matrix, $A$, its determinant can be written as $\vert A\vert$ or $\det(A)$.

A determinant can only be calculated from a square matrix.

The determinant of a $1\times1$ matrix:

\begin{displaymath}\vert a\vert = a\end{displaymath}

The determinant of a $2\times2$ matrix:

\begin{displaymath}\left\vert\begin{array}{cc}
a & b \\
c & d \end{array}\right\vert = ad - bc\end{displaymath}

The determinant of a $3\times3$ matrix:

\begin{displaymath}\left\vert\begin{array}{ccc}
a & b & c \\
d & e & f \\
g & ...
...
g & h \end{array}\right\vert =
a(ei-fh) - b(di-fg) + c(dh-eg)\end{displaymath}

More generally, to calculate the determinant of a matrix, for example:

\begin{displaymath}\left\vert\begin{array}{cccc}
a & b & c & d \\
e & f & g & h \\
i & j & k & l \\
m & n & o & p \end{array}\right\vert\end{displaymath}

Take the first row:

\begin{displaymath}a + b + c + d\end{displaymath}

Make every second element negative:

\begin{displaymath}a - b + c - d\end{displaymath}

For each element, ignore the row and column it's in:

\begin{displaymath}
\left\vert\begin{array}{cccc}
\textcolor{Blue}{a} & \textcol...
...}{l} \\
m & n & o & \textcolor{Red}{p} \end{array}\right\vert
\end{displaymath}

And multiply it by the determinant of the remaining elements:

\begin{displaymath}
a \left\vert\begin{array}{ccc}
f & g & h \\
j & k & l \\
n...
...}
e & f & g \\
i & j & k \\
m & n & o \end{array}\right\vert
\end{displaymath}

Then recursively decompose the resulting determinants until you're left with the trivial case of a single-element matrix.

(the above method applies to $2\times2$ and $3\times3$ matrices, but it helps to have a bigger matrix for demonstration)

Determinants have the property that:

\begin{displaymath}\left\vert AB\right\vert = \left\vert A\right\vert\left\vert B\right\vert\end{displaymath}

Copyright © 2005 Emil Mikulic.