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Determinant - Example

Determinant - Example: Encyclopedia II - Determinant - Example

Suppose we want to compute the determinant of We can go ahead and use the Leibniz formula directly: Alternatively, we can use Laplace's formula to expand the determinant along a row or column. It is best to choose a row or column with many zeros, so we will expand along the second column:

See also:

Determinant, Determinant - Determinants of 2-by-2 matrices, Determinant - Applications, Determinant - General definition and computation, Determinant - Example, Determinant - Properties, Determinant - Derivative, Determinant - Generalizations and related functions, Determinant - Algorithmic implementation, Determinant - History

Determinant, Determinant - Algorithmic implementation, Determinant - Applications, Determinant - Derivative, Determinant - Determinants of 2-by-2 matrices, Determinant - Example, Determinant - General definition and computation, Determinant - Generalizations and related functions, Determinant - History, Determinant - Properties

Determinant: Encyclopedia II - Determinant - Example

Determinant - Example

Suppose we want to compute the determinant of

We can go ahead and use the Leibniz formula directly:

Alternatively, we can use Laplace's formula to expand the determinant along a row or column. It is best to choose a row or column with many zeros, so we will expand along the second column:

A third way (and the method of choice for larger matrices) would involve the Gauss algorithm. When doing computations by hand, one can often shorten things dramatically by smartly adding multiples of columns or rows to other columns or rows; this doesn't change the value of the determinant, but may create zero entries which simplifies the subsequent calculations. In our example, adding the second column to the first one is especially useful:

and this determinant can be quickly expanded along the first column:

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

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