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If we multiply the first row by – 3, and add it to the second row, we get,
And once again, the same solution is maintained.
Now that we understand how the three row operations work, it is time to introduce the Gauss-Jordan method to solve systems of linear equations.
As mentioned earlier, the Gauss-Jordan method starts out with an augmented matrix, and by a series of row operations ends up with a matrix that is in the reduced row echelon form . A matrix is in the reduced row echelon form if the first nonzero entry in each row is a 1, and the columns containing these 1's have all other entries as zeros. The reduced row echelon form also requires that the leading entry in each row be to the right of the leading entry in the row above it, and the rows containing all zeros be moved down to the bottom.
We state the Gauss-Jordan method as follows.
Solve the following system by the Gauss-Jordan method.
We write the augmented matrix.
We want a 1 in row one, column one. This can be obtained by dividing the first row by 2, or interchanging the second row with the first. Interchanging the rows is a better choice because that way we avoid fractions.
we interchanged row 1(R1) and row 2(R2)
We need to make all other entries zeros in column 1. To make the entry (2) a zero in row 2, column 1, we multiply row 1 by - 2 and add it to the second row. We get,
To make the entry (3) a zero in row 3, column 1, we multiply row 1 by –3 and add it to the third row. We get,
So far we have made a 1 in the left corner and all other entries zeros in that column. Now we move to the next diagonal entry, row 2, column 2. We need to make this entry(–3) a 1 and make all other entries in this column zeros. To make row 2, column 2 entry a 1, we divide the entire second row by –3.
Next, we make all other entries zeros in the second column.
We make the last diagonal entry a 1, by dividing row 3 by – 4.
Finally, we make all other entries zeros in column 3.
Clearly, the solution reads , , and .
Before we leave this section, we mention some terms we may need in the fourth chapter. The process of obtaining a 1 in a location, and then making all other entries zeros in that column, is called pivoting . The number that is made a 1 is called the pivot element , and the row that contains the pivot element is called the pivot row . We often multiply the pivot row by a number and add it to another row to obtain a zero in the latter. The row to which a multiple of pivot row is added is called the target row .
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