Question

Given the matrix a = -1 3 4 1 0 3 -2 2 -1, compute a⁻¹ using the method of an augmented matrix and elementary row operations that was discussed in class?

Answer 1

To compute the inverse of a matrix using the method of an augmented matrix and elementary row operations, follow these steps: create an augmented matrix, perform row operations to transform the left side into the identity matrix, and perform the same row operations on the right side to obtain the inverse matrix.

Step-by-step explanation:

To compute the inverse of a matrix using the method of an augmented matrix and elementary row operations, follow these steps:

1. Create an augmented matrix by placing the given matrix and the identity matrix side by side. For the given matrix 'a', the augmented matrix would be:
2. [ -1 3 4 | 1 0 0 ]
3. 
   
4. [ 1 0 3 | 0 1 0 ]
5. 
   
6. [ -2 2 -1 | 0 0 1 ]
7. Perform row operations to transform the left side of the augmented matrix into the identity matrix. Use elementary row operations such as swapping rows, multiplying a row by a non-zero scalar, or adding a multiple of one row to another. Work through each row to create a diagonal of 1s on the left side.
8. Perform the same row operations on the right side of the augmented matrix to obtain the inverse of the given matrix. Once the left side of the augmented matrix is the identity matrix, the right side will be the inverse matrix. In this case, the inverse of matrix 'a' is:
9. [ 3 -11 -1 ]
10. 
    
11. [ -2 7 1 ]
12. 
    
13. [ 2 -7 -1 ]