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Mastering Linear Systems: Cramer’s Rule & Gaussian Methods

Learn linear system solving using Cramer’s Rule, Gaussian Elimination, Gauss-Jordan reduction with example problems. Master principles of Gauss elimination & homogenous solutions. Practice with homework exercises.

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Mastering Linear Systems: Cramer’s Rule & Gaussian Methods

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  1. Extra -2Review of Linear Systems By: Prof. Y. Peter Chiu 9 / 2011

  2. § . L 23 : Cramer’s Rule A‧X = B X = B =

  3. § . L 23 : Cramer’s Rule X = A-1B xi=

  4. Example 23-1 -2X1 + 3X2 - X3= 1 X1 + 2X2 - X3 = 4 X1= -2X1 - X2 + X3=-3 B→ X1= B→ B

  5. § . L 24 : If ≠ 0 Then ① A-1 exist ② Linear System has nontrivial solution. (非 0 解) ③ rank A = n ④ The rows (columns) of A are linearly independent.

  6. § . L 26 : Gauss- Jordan reduction

  7. § . L 25 : Gaussian Elimination高斯消去法 上三角(upper triangular)

  8. § . L 27 : Homework ＃1 X1+ X2 + 2X3 =-1 X1- 2X2 + X3 =-5 3X1+ X2 + X3 = 3 (a) Using Gaussian Elimination method to find solution. (b) Using Gauss-Jordan reduction method. (c) Using Cramer’s rule ＃2 2X1+ 4X2 + 6X3 = 2 X1+ 2X3 = 0 2X1+ 3X2 - X3 =-5 Using Cramer’s rule to solve it.

  9. § . L 27 : Homework ＃3 Solve 3 X1- X2= 3 ＃4 Solve2X1+ X2 +3X3 = 2 X1+ X3= 1 ＃5SolveX1+ 2X2 +3X3 = 6 4X1+ X3 = 4 2X1+ 4X2 +6X3 = 11

  10. The End

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