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

If  tex2html_wrap_inline1864 , 		 then det tex2html_wrap_inline1866 ,

If tex2html_wrap_inline1860 , then det tex2html_wrap_inline1862 .

For a square matrix A of dimensions tex2html_wrap_inline1818 , the determinant can be obtained as follows. First, define tex2html_wrap_inline1876 as the matrix of dimensions tex2html_wrap_inline1878 obtained from A by deleting row 1 and column j. Then

det tex2html_wrap_inline1866 det tex2html_wrap_inline1888 det tex2html_wrap_inline1890 det tex2html_wrap_inline1892 det tex2html_wrap_inline1894

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

For example, if tex2html_wrap_inline1898 , then

det tex2html_wrap_inline1900

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 13:24:14 EDT 1998