16.6 Rule of Sarrus

#Type/Definition #Topic/Linear_Algebra #Type/Example

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: 16.5 Permutation Definition of Determinant

Properties: N/A
Sufficiencies: N/A
Questions:

How do you compute a determinant of a matrix that is bigger than $3 \times 3$ ? #question

Rule of Sarrus

The rule of Sarrus is a mnemonic device for computing the determinant of a $3 \times 3$ matrix.
!300
The determinant of the three columns on the left is the sum of the products along the down-right diagonals minus the sum of the products along the up-right diagonals.

Remark. Another way to think of Sarrus' rule is to imagine that the matrix is wrapped around a cylinder, such that the right and left edges are joined.
!300
$⧫$

Rule of Sarrus

Consider a matrix $A \in R^{3 \times 3}$ .

A = [\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}]

\begin{proof}[Solution.]
We can compute the determinant by following the steps of the Rule of Sarrus.

\begin{aligned} det (A) & = \begin{array}{ccccc} a_{11} & a_{12} & a_{13} & a_{11} & a_{12} \\ a_{21} & a_{22} & a_{23} & a_{21} & a_{22} \\ a_{31} & a_{32} & a_{33} & a_{31} & a_{32} \end{array} \\ = a_{11} a_{22} a_{33} + a_{12} a_{23} a_{31} + a_{13} a_{21} a_{32} \\ - a_{13} a_{22} a_{31} - a_{11} a_{23} a_{32} - a_{12} a_{21} a_{33} \end{aligned}

\end{proof}