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Understanding Inverses of Matrices: Elementary Row Operations and Gaussian Elimination

This resource provides an in-depth exploration of matrix inverses, focusing on the definition and properties of inversible matrices. Key topics include the use of elementary row operations and Gauss-Jordan elimination to find inverses for both 2x2 and n x n matrices. It covers the criteria for matrix invertibility, the formulation of inverses, and alternative methods for solving systems of linear equations, which can enhance understanding and problem-solving skills in calculus and linear algebra.

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Understanding Inverses of Matrices: Elementary Row Operations and Gaussian Elimination

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  1. MAT 1234Calculus I 9.6 Inverses of Matrices http://myhome.spu.edu/lauw

  2. HW • WebAssign 9.6 • Wednesday/Monday Quiz: 9.4, 9.5, 9.6

  3. Recall (9.4) • Introduce Elementary Row Operations • Gauss-Jordan Eliminations

  4. Recall (9.5)

  5. Preview • The definition of the Inverse of a matrix. • Formula of the inverse for 2x2 matrices. • Use row operations to find the inverse of nxn matrices. • A second method of solving system of linear equations.

  6. Recall Identity Matrix nxn Square Matrix

  7. Inverse Matrix • Let A be a square matrix. Then the inverse for A is a square matrix A-1 of the same size as A such that AA-1 = I = A-1A

  8. Inverse Matrix • Let A be a square matrix. Then the inverse for A is a square matrix A-1 of the same size as A such that AA-1 = I = A-1A • If such inverse A-1 exists, then the matrix A is said to be invertible.

  9. Inverse Matrix • Let A be a square matrix. Then the inverse for A is a square matrix A-1 of the same size as A such that AA-1 = I = A-1A • If such inverse A-1 exists, then the matrix A is said to be invertible. • We will focus only on invertible matrices (to save time).

  10. Comparison

  11. Example 1 (a)

  12. How to Find Inverses? • 2x2 -> Formula • nxn -> Row Operations

  13. Inverse of a 2x2 Matrix

  14. Example 1 (b)

  15. Matrix Equations

  16. Example 1 (c) Use matrix inverse to solve

  17. Inverse of an 3x3 Matrix (Same for nxn matrices) Given matrix A, we set up the following matrix

  18. Inverse of an 3x3 Matrix (Same for nxn matrices) Given matrix A, Use row operations to get to the second matrix. A-1(if exists) is the matrix on the right half.

  19. Idea The row operations “effectively” multiply the matrix A by A-1to get I.

  20. Example 2 (a) Find the inverse of

  21. Example 2 (a)

  22. Example 2 (b) (=9.4 Example 2)

  23. Example 2 (b) (=9.4 Example 2)

  24. Q&A We can use two methods to solve a system of equations. (a) Gauss-Jordan Elimination (b) Matrix Inverse Q: Why use (b) when (a) is easier? A:

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