Understanding Linear Equations and Matrix Operations

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Explore the concepts of linear equations, matrix forms, determinants, and finding solutions for variables like x1, x2, x3. Learn about Cramer's Rules, Adjoint Matrix, and calculating the inverse of a matrix through examples and formulas.


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  1. Example : Find x1and x2from these equation : Solution :

  2. General form : Where : a1, a2, a3.. anand b are constantas x1, x2, x3.. xnare variables

  3. Linear Equation in matrix form If we have some equations : Then, we can write :

  4. General form : Where : Or, we can write :

  5. Example : Find x1and x2from these equation : Solution : Find A-1....

  6. A-1 : Formula :

  7. Cramers rules Assume : Determinants :

  8. x1 : x2 : x3 :

  9. Example : Find x1 and x2 from these equation : Solution :

  10. x1 and x2 : Proof :

  11. Find x1, x2 and x3 from these equations :

  12. Find Determinants :

  13. Find x1, x2 and x3 :

  14. Adjoint Matrix Adjoint matrix of a square matrix is the transpose of the matrix formed by cofactors of elements of determinant |A| How to calculate adjoint : Calculate minor matrix for each element of matrix Make cofactor matrix cofactor is a sign minor, denoted by : Cij = (-1)ij . Mij Change to Transpose matrix.

  15. Example Find inverse for A : Calculate |A| : =(1.5.3 + 2.0.2 + 3.0.4) (3.5.2 +1.0.4 + 2.0.3) = 15 30 = - 15

  16. Make a new matrix with minor and cofactor Transpose that matrix :

  17. Find x1, x2 and x3 from these equations :

  18. Matrix form : Formula :

  19. Determinants : Use minor cofactor :

  20. New matrix K : A-1 :

  21. Formula :

  22. Questions Find x1 and x2 from these equations : 2x1+ x2 4 = 0 x1 3x2+ 5 = 0 Find x, y and z from these equations : x + y + z 6 = 0 2x z + 1 = 0 x y + 2z 5 = 0

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