Properties of Determinants
Date: 10/23/2000 at 11:15:05 From: Nooni Subject: Solving determinants Dear Dr. Math, How do we solve determinants and how do we get them in echelon or row-echelon form? Are the steps we use based on certain things? Please give me an example to help me understand more.
Date: 10/23/2000 at 15:05:26 From: Doctor Rob Subject: Re: Solving determinants Thanks for writing to Ask Dr. Math, Nooni. You cannot "solve" a determinant. A determinant is an expression, and you cannot solve an expression, only evaluate it. You can solve equations, but a determinant is not an equation. I will assume you mean to evaluate a determinant. Properties of a determinant D: (a) If the corresponding rows and columns of D are interchanged (that is, D is transposed), the value of D is unchanged. (b) If any two rows (or columns) of D are interchanged, the value of D is changed to -D. (c) If any two rows (or columns) of D are alike, then D = 0. (d) If each element of a row (or column) of D is multiplied by m, then the value of D is changed to m*D. (e) If to each element of a row (or column) of D is added m times the corresponding element in another row (or column), the value of D is unchanged. The process of reducing a determinant to row-echelon form is a series of applications of these rules, especially (b), (d), and (e), using rows, to create a determinant whose value is a known multiple of the value of D, and which has zeroes below the main diagonal. Then the value of this new determinant is the product of the entries on the main diagonal. From that, you can figure out the value of D. Example. Evaluate the following determinant: | 2 1 -2| D = | 1 1 1| |-1 -2 3| Add -1/2 times the first row to the second row (Property (e)): | 2 1 -2| D = | 0 1/2 2| |-1 -2 3| Add 1/2 times the first row to the third row ((e) again): |2 1 -2| D = |0 1/2 2| |0 -3/2 2| Add 3 times the second row to the third row ((e) again): |2 1 -2| D = |0 1/2 2| |0 0 8| Now D is in row-echelon form. When it is in this form, the value of D is the product of the entries on the main diagonal, so D = 2*(1/2)*8 = 8 - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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