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

If  tex2html_wrap_inline5850 , 		 then det tex2html_wrap_inline5852 ,

If tex2html_wrap_inline5846 , then det tex2html_wrap_inline5848 .

For a square matrix A of dimensions tex2html_wrap_inline5804 , the determinant can be obtained as follows. First, define tex2html_wrap_inline5862 as the matrix of dimensions tex2html_wrap_inline5864 obtained from A by deleting row 1 and column j. Then

det tex2html_wrap_inline5852 det tex2html_wrap_inline5874 det tex2html_wrap_inline5876 det tex2html_wrap_inline5878 det tex2html_wrap_inline5880

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

For example, if tex2html_wrap_inline5884 , then

det tex2html_wrap_inline5886

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

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 16:30:59 EDT 1998