Matrix Row Operations and Inverses Help
Introduction to Matrix Row Operations and Inverses
We will use row operations to solve systems of equations and to find the multiplicative inverse of a matrix. These operations are similar to the elimination by addition method studied in Chapter 10. We will add two rows at a time (or some multiple of the rows) to make a particular entry 0. For example in the matrix we might want to change the entry with a 4 in it to 0. To do so, we can multiply the first row (Row 1) by −4 and add it to the second row (Row 2).
−4 Row 1 = −4(1 −3 2) = −4 12 −8
Using Row 2 and Row 3, change the entry with a 3 in it on the second row to 0.
When adding the rows together, we need the last entry in each column to be opposites. If we multiply Row 2 by −4 and Row 3 by 3, we will be adding −4(3) to 3(4) to get 0. Multiplying Row 2 by 4 and Row 3 by −3 also works.
Our first use for row operations is to find the inverse of a matrix (if it has one). If we multiply a matrix by its inverse, we get the corresponding identity matrix. For example,
To find the inverse of , we first need to write the augmented matrix. An augmented matrix for this method has the original matrix on the left and the identity matrix on the right.
We will use row operations to change the left half of the matrix to the 2 × 2 identity matrix. The inverse matrix will be the right half of the augmented matrix in Step 6.
Step 1 Use row operations to make the C entry a 0 for the new Row 2.
Step 2 Use row operations to make the B entry a 0 for the new Row 1 .
Step 3 Write the next matrix.
Step 4 Divide Row 1 by the A entry.
Step 5 Divide Row 2 by the D entry.
Step 6 Write the new matrix. The inverse matrix will be the right half of this matrix.
Step 1 We want to change −1, the C entry, to 0.
Step 2 We want to change −2, the B entry, to 0.
Step 4 This step is not necessary because dividing Row 1 by 1, the A entry, will not change any of its entries.
Step 5 Divide Row 2 by 2, the D entry.
The inverse matrix is
Finding the inverse of a 3 × 3 matrix takes a few more steps. Again, we will begin by writing the augmented matrix.
We will use row operations to turn the left half of the augmented matrix into the 3×3 identity matrix. There are many methods for getting from the first matrix to the last. The method outlined below will always work, assuming the matrix has an inverse.
Step 1 Use Row 1 and Row 2 to make the D entry to 0 for new Row 2.
Step 2 Use Row 1 and Row 3 to make the G entry to 0 for new Row 3.
Step 3 Write the next matrix.
Step 4 Use Row 1 and Row 2 to make the B entry a 0 for new Row 1.
Step 5 Use Row 2 and Row 3 to make the H entry a 0 for new Row 3.
Step 6 Write the next matrix.
Step 7 Use Row 1 and Row 3 to make the C entry a 0 for new Row 1.
Step 8 Use Row 2 and Row 3 to make the F entry a 0 for new Row 2.
Step 9 Write the next matrix.
Step 10 Divide Row 1 by A , Row 2 by E , and Row 3 by I. The inverse is the right half of the augmented matrix.
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