Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and chemometrics......click here for more.

## Matrix Determinants - Calculation of Order 2 and 3

The general approach how to calculate a matrix determinant is hard, requiring the calculation of many similar steps. Thus it is not recommended to calculate a determinant of matrices with an order higher than 3 without the help of a computer. For matrices of order 2 and 3 there are special rules which make it comparatively easy to determine the determinant:

Determinant of matrices of order 2

Let

 a11 a21 a12 a22

be an arbitrary matrix of order 2. Then its determinant is calculated as the product of the principal diagonal minus the product of the other diagonal, formally a11a22 - a12a21.

Determinant of matrices of order 3 (Sarrus' Rule)

Let

 a11 a21 a31 a12 a22 a32 a13 a23 a33

be an arbitrary matrix of order 3. Then its determinant is calculated as the sum of the product of all "extended" falling (including the principal) diagonals minus the sum of the product of all "extended" rising diagonals, formally (a11a22a33 + a21a32a13 + a31a12a23) - (a31a22a13 + a21a12a33 + a11a32a23). This rule is easier to understand when we color the relevant diagonals:

Example: determinant of a matrix of order 3

Let

Then

Please note that the rectangular, colored schemes do not denote actual matrices, but are only included to emphasize the rule of Sarrus.

Last Update: 2012-10-08