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Sarrus' method is a practical way to calculate the determinant of a matrix. This method is named after Pierre Frédéric Sarrus. To understand it, let's recall how to calculate the determinant of a matrix.
If we have a matrix:
Its determinant, or , is calculated as follows:
This is the difference between the product of the main diagonal elements () and the product of the secondary diagonal elements (). Sarrus' method adapts this principle for matrices.
Suppose we have a matrix :
The element is the element in the -th row and -th column.
Step : Copy the First Two Columns
Rewrite the first two columns of matrix to the right of the third column:
This helps us visualize the diagonals that will be multiplied.
Step : Calculate the Sum of the Products of the Positive Diagonals
Multiply the elements along the three diagonals from the top-left to the bottom-right. Sum these products, let's call it .
The first term is the product of the main diagonal. The second and third terms are products of parallel diagonals involving elements from the copied columns.
Step : Calculate the Sum of the Products of the Negative Diagonals
Multiply the elements along the three diagonals from the top-right to the bottom-left. Sum these products, let's call it .
The first term is the product of the secondary diagonal (anti-diagonal). The second and third terms are products of parallel diagonals involving elements from the copied columns, moving towards the bottom-left.
Step : Calculate the Final Determinant
The determinant of matrix , , is the difference between and :
Substitute the values of and :
Or, after distributing the negative sign:
To visualize this process, we can write:
Then, using Sarrus' Method, we expand the matrix and identify the multiplication paths:
Thus, the complete formula becomes:
Sarrus' method is only applicable to and matrices. For matrices of higher order (e.g., ), this method cannot be used. Other methods such as cofactor expansion or row reduction are required for such cases.
