Determining Inverse of 3x3 Matrices

Lecture Notes: Finding the Inverse of a 3x3 Matrix

Overview

The lecture demonstrates the step-by-step process to determine the inverse of a 3x3 matrix, using elementary row operations. The process involves converting the given matrix into an augmented matrix with the identity matrix and gradually transforming it to reveal the inverse.

Details of the Steps

1. Setting Up the Augmented Matrix

   • Given Matrix A:

     1  2 -1
     -2  0  1
     1  -1 0
     

   • Identity Matrix (I3):

     1 0 0
     0 1 0
     0 0 1
     

   • Augmented Matrix [A | I]:

     [1  2 -1 | 1 0 0]
     [-2 0  1 | 0 1 0]
     [1 -1  0 | 0 0 1]
     

2. Elementary Row Operations

   • Goal: Transform the left side of the augmented matrix to look like the identity matrix using elementary row operations.
   • The operations applied to the left must also be applied to the right to keep track of changes.

3. Operations and Transformations

   • Subtract Row 1 from Row 3;
   • Multiply Row 1 by 2 and add it to Row 2 to make the first element of Row 2 zero;
   • Multiply Row 2 by 3 and subtract 4 times Row 3, making modifications to cancel to zero;
   • Adjust Row 1 and Row 3 by adding them, followed by further adjustments among rows, particularly working to get zeros below and above leading ones;
   • Scale rows as necessary to simplify to identity form.

4. Verifying the Inversion

   • The resultant matrix on the right side after these transformations will be the inverse of Matrix A.
   • Multiply the original matrix A by the obtained inverse to verify if the product equals the identity matrix, confirming the correctness of the inverse.

5. Final Verification

   • Matrix calculations are carried out to confirm that the product of Matrix A and its derived inverse yields the identity matrix:

     I3 = 1 0 0
          0 1 0
          0 0 1
     

Conclusion

This systematic approach using augmented matrices and row operations allows for the accurate determination of inverses of 3x3 matrices. The verification step ensures the derived matrix is indeed the inverse. This method is fundamental in linear algebra and essential for solving systems of linear equations, among other applications.