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Determinant - General definition and computation

Determinant - General definition and computation: Encyclopedia II - Determinant - General definition and computation

Suppose is a square matrix. If A is a 1-by-1 matrix, then If A is a 2-by-2 matrix, then For a 3-by-3 matrix A, the formula is more complicated: For a general n-by-n matrix, the determinant was defined by Gottfried Leibniz with wha ...

Determinant: Encyclopedia II - Determinant - General definition and computation

Determinant - General definition and computation

Suppose is a square matrix.

If A is a 1-by-1 matrix, then

If A is a 2-by-2 matrix, then

For a 3-by-3 matrix A, the formula is more complicated:

For a general n-by-n matrix, the determinant was defined by Gottfried Leibniz with what is now known as the Leibniz formula:

The sum is computed over all permutations σ of the numbers {1,2,...,n} and sgn(σ) denotes the signature of the permutation σ: +1 if σ is an even permutation and −1 if it is odd (see even and odd permutations).

This formula contains n! (factorial) summands and is therefore impractical to use it to calculate determinants for large n.

In general, determinants can be computed with the Gauss algorithm using the following rules:

If A is a triangular matrix, i.e. whenever i > j, then
If B results from A by exchanging two rows or columns, then
If B results from A by multiplying one row or column with the number c, then
If B results from A by adding a multiple of one row to another row, or a multiple of one column to another column, then

Explicitly, starting out with some matrix, use the last three rules to convert it into a triangular matrix, then use the first rule to compute its determinant.

It is also possible to expand a determinant along a row or column using Laplace's formula, which is efficient for relatively small matrices. To do this along row i, say, we write

where the C_i,j represent the matrix cofactors, i.e. C_i,j is ( − 1)^{i + j} times the minor M_i,j, which is the determinant of the matrix that results from A by removing the i-th row and the j-th column.

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Adapted from the Wikipedia article "General definition and computation", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki

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