## Determinants

The determinant of a matrix , denoted , is only defined when is a square matrix, i.e., the number of rows is equal to the number of columns. If is an matrix then may be regarded as a function of , sending vectors to points.

If then it turns out the matrix , regarded as a function , will send the unit hypercube ( times) in to a parallelepiped (the -dimensional analog of a parallelogram). It is known that the absolute value of the determinant of measures the volume of that parallelpiped. For example, sends the vertices of the unit square to the points , , , . These points bound a parallelogram having volume .

If the determinant is easy to define:

The easiest constructive way to define the determinant of an arbitrary matrix is to use the Laplace cofactor expansion (along the row): for any , we have

 (5.2)

where is the `submatrix of ' obtained by omitting all the entries in the row or the column. The matrix is called the -minor of and is called the -cofactor.

The determinant has many remarkable properties. For example,

• You can factor an expression out of any row or column. For example,

• The Lagrange expansion holds for any row or column. For example, the middle column expansion for a matrix:

• The determinant, as a function of its rows or columns, is additive. For example,

For a proof, see any text on linear algebra (e.g., [JN]).

A square matrix is singular if . Otherwise, is called non-singular or invertible.

Example 5.5.1   Taking ,

This implies is singular. Indeed, the parallelepiped generated by , , , must be flat ( -dimensional, hence have 0 volume) since and .

An important fact about singular matrices, and one that we will use later, is the following.

Lemma 5.5.2   Suppose is an matrix with entries in . The following are equivalent:
• ,

• there is no non-zero vector such that , where 0 is the zero vector in ,

• exists.

For a proof, see any text on linear algebra.

We end this section with one last key property of determinants.

Lemma 5.5.3   If are any two matrices having entries in then .

More details on all this material can be found in any text book on linear algebra.

david joyner 2008-04-20