Determinants

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.

For our purpose, however, determinants will be needed mainly in our discussion of classical optimization, in conjunction with the material from the following section.