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# 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.

Michael A. Trick
Mon Aug 24 13:24:14 EDT 1998