Difference between revisions of "User:Temperal/The Problem Solver's Resource4"
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The determinant of a <math>2</math> by <math>2</math> (said to have order <math>2</math>) matrix <math>\left | | The determinant of a <math>2</math> by <math>2</math> (said to have order <math>2</math>) matrix <math>\left | | ||
===General Formula for the Determinant=== | ===General Formula for the Determinant=== | ||
− | Let <math>A</math> be a square matrix of order <math>n</math>. Write <math>A = a_{ij}</math>, where <math>a_{ij}</math> is the entry on the row <math>i</math> and the column <math>j</math>, for <math>i=1,\cdots,n</math> and <math>j=1,\cdots,n</math>. For any <math>i</math> and <math>j</math>, set <math>A_{ij}</math> (called the cofactors) to be the determinant of the square matrix of order <math>n-1 | + | Let <math>A</math> be a square matrix of order <math>n</math>. Write <math>A = a_{ij}</math>, where <math>a_{ij}</math> is the entry on the row <math>i</math> and the column <math>j</math>, for <math>i=1,\cdots,n</math> and <math>j=1,\cdots,n</math>. For any <math>i</math> and <math>j</math>, set <math>A_{ij}</math> (called the cofactors) to be the determinant of the square matrix of order <math>n-1</math> obtained from <math>A</math> by removing the row number <math>i</math> and the column number <math>j</math> multiplied by <math>(-1)^{i+j}</math>. Thus: |
<math>\det(A) = \sum_{j=1}^{j=n} a_{ij} A_{ij}</math> | <math>\det(A) = \sum_{j=1}^{j=n} a_{ij} A_{ij}</math> |
Revision as of 16:27, 8 December 2007
AlgebraThis is a collection of algebra laws and definitions. Obviously, there is WAY too much to cover here, but we'll try to give a good overview. Elementary AlgebraDefinitions
Factor TheoremIff a polynomial has roots , then , and are all factors of . Quadratic FormulaFor a quadratic of form , where are constants, the equation has roots Fundamental Theorems of Algebra
Third-degree and Quartic FormulasIf third-degree polynomial has roots , then: The three roots of a cubic polynomial equation are given implicitly by Quartic formulas are listed here. The general quintic equation (or an equation of even higher degree) does not have a formula. DeterminantsThe determinant of a by (said to have order ) matrix is . General Formula for the DeterminantLet be a square matrix of order . Write , where is the entry on the row and the column , for and . For any and , set (called the cofactors) to be the determinant of the square matrix of order obtained from by removing the row number and the column number multiplied by . Thus:
Cramer's LawConsider a set of three linear equations (i.e. polynomials of degree one) Let , , , , , and . This can be generalized to any number of linear equations. Abstract AlgebraIncomplete. Diophantine EquationsIncomplete. |