# How do you solve using gaussian elimination or gauss-jordan elimination, #x-2y-z=2#, #2x-y+z=4#, #-x+y-2z=-4#?

##### 1 Answer

#### Answer:

Perform elementary row operations to obtain an upper triangular matrix , and use this result to solve the equation.

#### Explanation:

First, we place the equation into matrix form, Ax=b:

Becomes:

Giving us:

From here, we will use elementary row operations to solve this. Our altered matrix, A', should be an upper triangular matrix; that is, one where the only non-zero entries are along the main diagonal (upper left corner to lower right) and above. Thus, there will be no non-zero entries below (or to the left) of the main diagonal.

Let us define

First, we will change entries *also* transform the corresponding b values; essentially at this point, we are adding and subtracting equations to/from one another, which by necessity will include adding and subtracting the answers of the original equations. We will begin by subtracting twice row 1 from row 2 (to reduce the first entry in row 2 to 0), and then we will add row 1 to row 3 (reducing the first entry in row 3 to 0). These new rows will be labelled as

And

At this point in our operations we should have:

Next, we wish to reduce entry

This yields our upper triangular matrix as:

From here, we note that row 3 multiplies out to

We may then multiply out row 2 (i.e. multiplying the row by our x column vector to get the appropriate result from our b vector) and recall that

Performing a similar trick with our first row we get:

Since