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Table of Contents Math Background Matrices Matrix Determinants - Calculation of Order 2 and 3 | |
See also: matrix determinant, rank of a matrix |
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
a_{11}
a_{21} |
a_{12}
a_{22} |
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 a_{11}a_{22 }- a_{12}a_{21}.
Determinant of matrices of order 3 (Sarrus' Rule)
a_{11}
a_{21} a_{31} |
a_{12}
a_{22} a_{32} |
a_{13}
a_{23} a_{33} |
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 (a_{11}a_{22}a_{33} + a_{21}a_{32}a_{13} + a_{31}a_{12}a_{23}) - (a_{31}a_{22}a_{13} + a_{21}a_{12}a_{33} + a_{11}a_{32}a_{23}). This rule is easier to understand when we color the relevant diagonals:
Example: determinant of a matrix of order 3
Let
Then
Last Update: 2006-Jän-17