rule of Sarrus

Rule of Sarrus

Sarrus_ABC_red_blue_solid_dashed.svg.png

Rule of Sarrus is a simple formula for computing the determinant of a \[3\times 3\] matrix.

Consider a \[3\times 3\] matrix, \[M=\begin{pmatrix} a&b&c\\d&e&f\\g&h&i \end{pmatrix}\], then \[\det(M)=aei+bfg+cdh-ceg-bdi-afh\]. One can also apply this pattern to a \[2\times 2\] matrices, such that if \[N=\begin{pmatrix} a&b\\c&d \end{pmatrix}\], then \[\det(N)=ad-bc\].

Proof

Let \[M=\begin{pmatrix} a&b&c\\d&e&f\\g&h&i \end{pmatrix}\]. Using Laplace expansion,

\begin{align*} \det(M)&=a\cdot\det\begin{pmatrix} e&f\\h&i \end{pmatrix}-b\cdot\det\begin{pmatrix} d&f\\g&i \end{pmatrix}+c\cdot\det\begin{pmatrix} d&e\\g&h \end{pmatrix}\\ &=a\cdot(ei-fh)-b\cdot(di-fg)+c\cdot(dh-eg)\\ &=aei-afh-bdi+bfg+cdh-ceg \end{align*}
index