To each square matrix,
we associate a number, called its *determinant*, defined as follows:

If , then det ,If , then det .

For a square matrix *A* of dimensions , the determinant
can be obtained as follows. First,
define as the matrix of dimensions
obtained from *A* by deleting row 1 and
column *j*. Then

det det det det det

Note that, in this formula, the signs alternate between + and -.

For example, if , then

det

Determinants have several interesting properties. For example,
the following statements are equivalent for a square matrix *A*:

- det(
*A*) =0, -
*A*has no inverse, i.e.*A*is singular, - the columns of
*A*form a set of linearly dependent vectors, - the rows of
*A*form a set of linearly dependent vectors.

Mon Aug 24 13:24:14 EDT 1998