Using a row operation, compute the inverse of the matrix A= ( 1 1 1
0 2 3
5 5 1)
To compute the inverse of matrix A using row operations, we start by augmenting A with the identity matrix of the same size.
Augmented Matrix:
[ 1 1 1 | 1 0 0 ]
[ 0 2 3 | 0 1 0 ]
[ 5 5 1 | 0 0 1 ]
Next, we perform row operations to transform the left-hand side of the augmented matrix into the identity matrix. The right-hand side will then become the inverse of A.
1. R2 = R2 - 2R1:
[ 1 1 1 | 1 0 0 ]
[ 0 0 1 | -2 1 0 ]
[ 5 5 1 | 0 0 1 ]
2. R3 = R3 - 5R1:
[ 1 1 1 | 1 0 0 ]
[ 0 0 1 | -2 1 0 ]
[ 0 0 -4 | -5 0 1 ]
3. R3 = -R3/4:
[ 1 1 1 | 1 0 0 ]
[ 0 0 1 | -2 1 0 ]
[ 0 0 1 | 5/4 0 -1/4 ]
4. R1 = R1 - R3 and R2 = R2 - R3:
[ 1 1 0 | -1/4 0 1/4 ]
[ 0 0 0 | -3/4 1/4 1/4 ]
[ 0 0 1 | 5/4 0 -1/4 ]
5. R2 = -4/3R2:
[ 1 1 0 | -1/4 0 1/4 ]
[ 0 0 0 | 1 -1/3 -1/3 ]
[ 0 0 1 | 5/4 0 -1/4 ]
6. R1 = R1 - R2:
[ 1 1 0 | -5/12 1/3 5/12 ]
[ 0 0 0 | 1 -1/3 -1/3 ]
[ 0 0 1 | 5/4 0 -1/4 ]
7. R1 = -R1 and R2 = -R2:
[ -1 -1 0 | 5/12 -1/3 -5/12 ]
[ 0 0 0 | -1 1/3 1/3 ]
[ 0 0 1 | 5/4 0 -1/4 ]
8. R1 = -1/2R1:
[ 1 1 0 | -5/12 1/3 5/12 ]
[ 0 0 0 | -1 1/3 1/3 ]
[ 0 0 1 | 5/4 0 -1/4 ]
9. R1 = R1 + R3 and R2 = R2 - R3:
[ 1 1 0 | 0 1/3 3/6 ]
[ 0 0 0 | -1 1/3 1/3 ]
[ 0 0 1 | 5/4 0 -1/4 ]
10. R1 = R1 - R2:
[ 1 1 0 | 1 0 1/6 ]
[ 0 0 0 | -1 1/3 1/3 ]
[ 0 0 1 | 5/4 0 -1/4 ]
Therefore, the inverse of matrix A is:
[ 1 0 1/6 ]
[ -1 1/3 1/3 ]
[ 5/4 0 -1/4 ]
To compute the inverse of a matrix using row operations, we will first augment the given matrix A with the identity matrix of the same size. Then, we will perform row operations on the augmented matrix until the original matrix is converted to the identity matrix. The inverse matrix will then be the augmented part of the resulting matrix.
1. Augment the matrix A with the identity matrix:
A = [1 1 1 | 1 0 0]
[0 2 3 | 0 1 0]
[5 5 1 | 0 0 1]
2. Apply row operations to convert the left part of the augmented matrix to the identity matrix:
R2 = R2/2
R3 = R3 - 5R1
A = [1 1 1 | 1 0 0]
[0 1 3/2 | 0 1/2 0]
[0 0 -4 | -5 0 1]
3. Multiply the third row by -1/4 to convert the leading entry in the third row to 1:
R3 = -1/4 * R3
A = [1 1 1 | 1 0 0]
[0 1 3/2 | 0 1/2 0]
[0 0 1 | 5/4 0 -1/4]
4. Apply row operations to convert the other entries in the third column to zeros:
R2 = R2 - (3/2) * R3
R1 = R1 - R3
A = [1 1 0 | -1/4 0 1/4]
[0 1 0 | -15/8 1/2 3/8]
[0 0 1 | 5/4 0 -1/4]
5. Apply row operations to convert the other entries in the second column to zeros:
R1 = R1 - R2
A = [1 0 0 | 3 0 -1]
[0 1 0 | -15/8 1/2 3/8]
[0 0 1 | 5/4 0 -1/4]
The resulting matrix on the right side is the inverse of matrix A:
A^-1 = [3 0 -1]
[-15/8 1/2 3/8]
[5/4 0 -1/4]