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determinants

# determinants - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 200 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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det(A), 1A.J. - it.' a e i + b f g + d h c - c e g - b d i - f h a . I I IAl D-1. IAl IAl 0 IAl c, IAl (I), + 1 Laplace (1). here D. Determinants Given a square array A of numbers, we associate with it a number called the determinant of A, and written either or For 2 x 2 and 3 x 3 arrays, the number is defined by a b c (1) = ad - bc; = d e f g h i Do not memorize these as formulas learn instead the patterns which give the terms. The 2 x 2 case is easy: the product of the elements on one diagonal (the " main diagonal " ), minus the product of the elements on the other (the " antidiagonal " ). For the 3 x 3 case, three products get the + sign: those formed from the main diagonal, or having two factors parallel to the main diagonal. The other three get a negative sign: those from the antidiagonal, or having two factors parallel to Try the following example on your own, then check your work against the solution. 1 - 2 1 Example 1.1 Evaluate - 1 3 2 using 2 - 1 4 Solution. Using the same order as in we get 12 ( - 8) + - 6 - 8 - ( - 2) = - 7 . Important facts about : is multiplied by - 1 if we interchange two rows or two columns. D - 2. = if one row or column is all zero, or if two rows or two columns are the same. D - 3. is multiplied by if every element of some row or column is multiplied by c. of another row (resp. column). D - 4. The value of is unchanged if we add to one row (or column) a constant multiple All of these facts are easy to check for 2 x 2 determinants from the formula (1); from this, their truth also for 3 x 3 determinants will follow from the expansion.
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