Lecture 24: Introductory Linear Algebra - Computing the Inverse of a Matrix
Introduction
Presenter: Nathan Johnston
Objective: Learn how to compute the inverse of a matrix.
Previous classes covered:
Determining if a matrix has an inverse.
The relationship between a matrix and its reduced row echelon form.
Key Theorems and Concepts
Theorem: Invertibility and Row Echelon Form
A matrix is invertible if its reduced row echelon form (RREF) is the identity matrix.
If you can row reduce the augment of a matrix A with an identity matrix to the identity matrix augmented with another matrix, the right-hand side is the inverse.
Note: Inverses must satisfy both A inverse times A equals identity and A times A inverse equals identity.
Steps to Find the Inverse of a Matrix
Augment the matrix with an identity matrix.
Row reduce to achieve the identity matrix on the left.
If successful, the right matrix is the inverse.
If not, the matrix is not invertible and the right side is irrelevant.
Example 1: Finding a 2x2 Matrix Inverse
Augment matrix A with an identity matrix.
Perform row operations to achieve identity on the left.
The matrix on the right is the inverse.
Verify by multiplying A and A inverse to ensure it results in the identity matrix.
Solving Linear Systems Using Inverses
Example: Solve x + 2y = 3 and 3x + 4y = 5 using matrix inverses.
Express system as matrix equation: AX = B.
Use inverse of A to find X: X = A inverse times B.
Solution: x = -1, y = 2.
Example 2: 3x3 Matrix
Methodology remains the same: augment with identity, row reduce.
Determine if the matrix is invertible and find inverse if it exists.
Successfully row reduce to identity confirms invertibility.
Example 3: Non-Invertible Matrix
Attempt to row reduce 3x3 matrix.
Encounter a zero row indicates the matrix is not invertible.
Zero row on left means reduced row echelon form is not identity.
Conclusion
Next Class: Further theorems on invertibility and explicit formula for 2x2 matrix inverses.
Goal is to avoid Gaussian elimination for 2x2 matrices.
Note: Additional verification steps are beyond the necessary procedure for finding inverses but are useful for checking work.