Matrix Determinant

Given a matrix, , 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:

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