Example of Gaussian Elimination Applied to a Redundant System of Linear Equations

Use Gaussian elimination to put this system of equations into triangular echelon form and solve it if possible:

Solution: Perform this sequence of E.R.O.'s on the augmented matrix. Set the pivot column to column 1. There is already a 1 in the pivot position, so proceed to get 0's below the pivot:

Now, set the pivot column to the second column. First, get a 1 in the diagonal position:

Next, get a 0 in the position below the pivot:

Now, set the pivot column to the third column. The first thing to do is to get a 1 in the diagonal position, but there is no way to do this. In fact this matrix is already in triangular echelon form and represents:

This system of equations can't be solved by back-substitution because we have no value for z. The last equation merely states that 0=0. There is no unique solution because z can take on any value.

In general, one or more rows of zeros at the bottom of an augmented matrix that has been put into triangular echelon form indicates a redundant system of equations.

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