Augmented Matrix Notation And Elementary Row Operations - Ximera

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Consider the linear system Our goal is to use elementary row operations to transform this system into an equivalent system of the form We have to keep in mind that given an arbitrary system, an equivalent system of this form may not exist (we will talk a lot more about this later), but it does exist in this case, and we would like to find a more efficient way of finding it than having to write and rewrite our equations at each step.

In this problem, we prompt you to perform elementary row operations and ask you to fill in the coefficients in the resulting equations.

If we drop all of the zero terms, we have: Now we see that is the solution.

Observe that throughout the entire process, variables , , and remained in place; only the coefficients in front of the variables and the entries on the right changed. Let’s try to recreate this process without writing down the variables. We can capture the original system in (eq:sys20originalsystem1) as follows:

This array is called an augmented matrix. The side to the left of the vertical bar is called the coefficient matrix, while the side to the right of the vertical bar corresponds to the constants on the right side of the system.

We can capture all of the elementary row operations we performed earlier as follows:

The last augmented matrix corresponds to systems in (eq:sys20rref1) and (eq:sys20rrefnozeros), and we can easily see the solution.

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