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Mastering Inverse Matrices for Equation Solutions

Learn how to find and use inverse matrices to solve systems of equations efficiently. This alternative method simplifies solving equations and finding solutions. Explore the process step by step with examples and shortcuts.

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Mastering Inverse Matrices for Equation Solutions

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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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