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8.3 Another Way of Solving a System of Equations

8.3 Another Way of Solving a System of Equations. Objectives: 1.) Learn to find the inverse m atrix 2.) Use the inverse m atrix to a system of equations. Consider this. Let A= Y= B=. Find Y if A + Y = B. Consider this. Let A= Y= B=. Find Y if AY = B.

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8.3 Another Way of Solving a System of Equations

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  1. 8.3 Another Way of Solving a System of Equations Objectives: 1.) Learn to find the inverse matrix 2.) Use the inverse matrix to a system of equations

  2. Consider this Let A= Y= B= Find Y if A + Y = B

  3. Consider this Let A= Y= B= Find Y if AY = B

  4. There is no division operation on matrices

  5. Alternative Form for Solving a System of Equations Using the Inverse Matrix New Notation Let A be the cofficient matrix Let X be the variable matrix Let B be the solution matrix Thus, AX= B

  6. Coefficient Matrix (A) • A matrix whose real entries are the coefficients from a system of equations

  7. Variable Matrix (X) • A column matrix of the unknown variables

  8. Solution Matrix • A column matrix whose entries are the solutions of the system of equations

  9. Identity Matrix • A square matrix with a diagonal of 1s and all other entries are zeros • RREF Form • Notation: I

  10. Characteristic of the Identity Matrix • When a matrix is multiplied by the identity, you get the same matrix; AI= A

  11. Example

  12. Inverse Matrix • Let A be a square matrix, then A-1 is the inverse matrix if AA-1 = I = A-1A

  13. Example • A = B= Thus B can be notated A-1 because it is the inverse of A.

  14. Finding the Inverse Matrix (The original matrix needs to be square!) 1.) Write the augmented matrix with [A:I] (The coefficient matrix and the identity matrix side by side 2.) Do proper row reductions to both A and I until A is in rref form (It has become an identity matrix itself 3.) The change in I is the inverse matrix of A, A-1 *** If you get a row of full zeros, the inverse does not exist****

  15. Example Pg. 579 #22

  16. Example: Find the inverse matrix of

  17. How this helps us solve a system of equations. Example: Pg. 580 #53

  18. Shortcut for finding the inverse of a 2x2 • Pg. 577: If A is invertible if ad-bc≠0 There is no inverse if ad-bc=0

  19. A is invertible if ad-bc≠0

  20. Homework: 8.3 • Page 579 # 2; 5; 19-22; 39-47(odd); 53; 54; 60; 71

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