Find the inverse of the matrices if they exist. Use the algorithm introduced in this section.
1 2 -1
-4 -7 3
-2 -6 4
Don't know the algorithm.
Inverse exists if determinant is nonzero
D = 0 in this case, so no inverse exists
To find the inverse of a matrix, we'll use the algorithm introduced in this section. Let's proceed step-by-step:
Step 1: Write the given matrix and an identity matrix side by side.
1 2 -1 | 1 0 0
-4 -7 3 | 0 1 0
-2 -6 4 | 0 0 1
Step 2: Perform row operations to get the left side of the matrix in echelon form.
Row1 = Row1/1
Row2 = Row2 + 4*Row1
Row3 = Row3 + 2*Row1
1 2 -1 | 1 0 0
0 1 -1 | 4 1 0
0 -2 2 | 2 0 1
Step 3: Perform row operations to get the left side of the matrix in reduced row echelon form.
Row2 = Row2 + 2*Row3
1 2 -1 | 1 0 0
0 -3 3 | 8 1 2
0 -2 2 | 2 0 1
Step 4: Perform row operations to get the right side of the matrix in reduced row echelon form.
Row2 = Row2/(-3)
1 2 -1 | 1 0 0
0 1 -1 | -8 -1 -2
0 -2 2 | 2 0 1
Step 5: Perform row operations to get the left side of the matrix in reduced row echelon form.
Row3 = Row3 + 2*Row2
1 2 -1 | 1 0 0
0 1 -1 | -8 -1 -2
0 0 0 | -14 -2 -3
Although the matrix on the left side is not in reduced row echelon form, we can see that the final row (row 3) consists of all zeros, which indicates that the inverse of the given matrix does not exist.
Hence, the inverse of the given matrix does not exist.
To find the inverse of a matrix using the algorithm introduced in this section, follow these steps:
Step 1: Set up the augmented matrix
Combine the given matrix with the identity matrix of the same size. In this case, the given matrix is a 3x3 matrix, so we'll set up a 3x6 augmented matrix.
1 2 -1 | 1 0 0
-4 -7 3 | 0 1 0
-2 -6 4 | 0 0 1
Step 2: Perform row operations to transform the left side (given matrix) into the identity matrix
Use row operations such as scaling, swapping, or adding multiples of one row to another to transform the left side of the augmented matrix into the identity matrix. Perform the same row operations on the right side of the augmented matrix.
Perform the following row operations:
- Row 2 = 4 * Row 1 + Row 2
- Row 3 = 2 * Row 1 + Row 3
The augmented matrix becomes:
1 2 -1 | 1 0 0
0 1 -1 | 4 1 0
0 -2 2 | 2 0 1
- Row 3 = 2 * Row 2 + Row 3
The augmented matrix becomes:
1 2 -1 | 1 0 0
0 1 -1 | 4 1 0
0 0 0 | 10 -2 1
Step 3: Continue row operations to transform the left side into the identity matrix and the right side into the inverse matrix
Continue performing row operations on the augmented matrix:
- Row 3 = -1/2 * Row 2 + Row 3
The augmented matrix becomes:
1 2 -1 | 1 0 0
0 1 -1 | 4 1 0
0 0 0 | 1 -1/2 1
- Row 1 = Row 1 + Row 3
The augmented matrix becomes:
1 2 -1 | 2 -1/2 1
0 1 -1 | 4 1 0
0 0 0 | 1 -1/2 1
- Row 1 = Row 1 + 2 * Row 2
The augmented matrix becomes:
1 0 -3 | 10 -1/2 1
0 1 -1 | 4 1 0
0 0 0 | 1 -1/2 1
Step 4: Divide each row by the pivot element on the left side
Divide each row of the augmented matrix by its pivot element on the left side to make the left side the identity matrix.
Divide Row 1 by 1:
1 0 -3 | 10 -1/2 1
Divide Row 2 by 1:
0 1 -1 | 4 1 0
The augmented matrix becomes:
1 0 -3 | 10 -1/2 1
0 1 -1 | 4 1 0
0 0 0 | 1 -1/2 1
Step 5: The right side of the augmented matrix is the inverse matrix
The right side of the augmented matrix, in this case, is the inverse matrix. Therefore, the inverse of the given matrix is:
10 -1/2 1
4 1 0
1 -1/2 1
So, the inverse of the given matrix exists, and it is:
10 -1/2 1
4 1 0
1 -1/2 1