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Direct sustitution methods
 

Direct sustitution methods

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    Direct sustitution methods Direct sustitution methods Presentation Transcript

    • RUBEN DARIO ARISMENDI RUEDA
    • CHAPTER 4: ‘Direct Methods to solve lineal ecuation systems’
    • Types of Matrix. 1. RECTANGULAR:The Matrix that has different number of lines and columns mxn, Where ‘m’ is different of ‘n’. 2. Transpossition Matrix: Is the new matrix, that is obtain when the files and columns are changed.
    • 3.Void Matrix. When all the elements of the Matrix are cero. 4.Simetric Matrix. Is square matrix equal to the trnsposition matrix. 5.Diagonal Matrix. Is a square matriz with all the elements cero except the element of the main diagonal.
    • 6.Identical Matrix. Is a square Matrix that has all the elements cero except the element of the main diagonal that are 1. 7. Triangular. (Inferior or Superior)
    • 8. Inverse Matrix. A matrix is inverse when:
    • METHODS. SIMPLE GAUSS. This method has two main steps. a- Forward elimination: Consist in convert the Matrix into a Superior Triangular b- Backward sustitution: With the Inferior Matrix the last unkown value is obtained and then we have to start making substitutions.
    • We can change the order of the lines to make it easier to solve
    • F2-3f1 f3-5f1 F3-2f2
    • NOW WE START TO MAKE THE SUSTITUTIONS.
    • GAUSS-JORDAN. This method consist in treat the Matrix and convert it into a identical Matrix, and then is more easy to solve the system.
    • F1 *(1/2) (F1*-3)+f2 (F1*-5)+f3 (F2*(-2/13)) (F3*-2)
    • (F2*17)+f3 (F3*13/96) (F3*-1/2)+f1 (F3*-11/13)+f2
    • (F2*-3/2)+f1 x= 1 y= -1 z= 2 SOLUTION:
    • 3. Inverse Matrix. The inverse matrix could be used to solve a system as well. 2x + 4y + 3z = 6 y – z = - 4 3x + 5y + 7z =3 THE SYSTEM COULD BE WRITTEN LIKE A *x=B
    • THE INVERSE x= 25 y= - 8 z= -4 THE SOLUTION IS