Fundamentals part 2

Fundamentals part 2 - Determinant I I determinant of a...

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Determinant determinant of a matrix is a scalar value usually only need 2x2 and 3x3 matrix determinants the determinant of M is | M | 21 12 00 22 10 01 20 11 02 21 10 02 20 12 01 22 11 00 21 20 11 10 02 22 20 12 10 01 22 21 12 11 00 22 21 20 12 11 10 02 01 00 m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m M + + = + = = Determinant Properties for an n x n matrix M M M s sM N M MN M M T n = = = = 1 1

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Inverse exists only if determinant is nonzero multiplicative inverse properties computing inverse? Cramer’s rule (we’ll see this soon) Gaussian elimination and other methods I M M MM = = 1 1 T T M M M N MN ) ( ) ( ) ( 1 1 1 1 1 = = Cofactor need this for Cramer’s rule cofactor of matrix element m ij is (-1) i+j times determinant of the matrix obtained by deleting row i and column j from M example (adapted from Hill A2.1.5) ú ú ú ú û ù ê ê ê ê ë é = 1 0 0 0 0 7 5 0 0 4 1 8 0 6 0 2 M

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