Ashish Aziz

Give the complete solution to the system of equations \

Solution

The augmented matrix of this system is \\] In order to find the solution to this system, we will carry the augmented matrix to reduced row-echelon form, using Reduced Row-Echelon Form Algorithm. The first column is the first pivot column. We want to use row operations to create zeros beneath the first entry in this column, which is in the first pivot position. Replace the third row with \(2\) times the first row added to \(3\) times the third row. This gives

\\]

Now, we have created zeros beneath the \(3\) in the first column, so we move on to the second pivot column (which is the second column) and repeat the procedure. Take \(-1\) times the second row and add to the third row. \\] The entry below the pivot position in the second column is now a zero. Notice that we have no more pivot columns because we have only two leading entries.

At this stage, we also want the leading entries to be equal to one. To do so, divide the first row by \(3\). \\]

This matrix is now in row-echelon form.

Let’s continue with row operations until the matrix is in reduced row-echelon form. This involves creating zeros above the pivot positions in each pivot column. This requires only one step, which is to add \(\frac{1}{3}\) times the second row to the first row. \\]

This is in reduced row-echelon form, which you should verify using Definition 11.3.4. The equations corresponding to this reduced row-echelon form are \ or \

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