Gauss-Jordan: Find The Augmented Matrix

Hey guys! Today, we're diving into the fascinating world of linear systems and how to solve them using the mighty Gauss-Jordan elimination. Specifically, we're going to tackle the question of how to represent a given linear system as an augmented matrix. This is a crucial first step in applying the Gauss-Jordan method, so let's get right to it!

What is Gauss-Jordan Elimination?

Before we jump into the specifics, let's briefly recap what Gauss-Jordan elimination actually is. At its core, Gauss-Jordan elimination is a systematic method for solving systems of linear equations. It's a powerful technique that allows us to find the values of the unknowns (variables) that satisfy all the equations in the system simultaneously. The beauty of this method lies in its structured approach, which makes it less prone to errors compared to manual substitution or elimination methods.

Gauss-Jordan elimination hinges on the concept of elementary row operations. These are operations that we can perform on the rows of a matrix without changing the underlying solution of the linear system. There are three primary elementary row operations:

1. Swapping two rows: This is as simple as it sounds – we can interchange the positions of any two rows in the matrix. This is useful for rearranging the equations or getting a leading 1 in the desired position.
2. Multiplying a row by a non-zero constant: We can multiply all the elements in a row by the same non-zero number. This is handy for creating leading 1s or eliminating coefficients.
3. Adding a multiple of one row to another row: This is the workhorse of the method. We can add a multiple of one row to another row, which allows us to eliminate variables strategically.

The main goal of Gauss-Jordan elimination is to transform the augmented matrix into what's called reduced row-echelon form. This form has a very specific structure:

• Leading 1s: Each non-zero row has a leading 1 (also called a pivot), which is the first non-zero entry in the row.
• Zeroes below leading 1s: All entries below each leading 1 are zeroes.
• Zeroes above leading 1s: All entries above each leading 1 are also zeroes.
• Leading 1s move rightwards: The leading 1 in any row is to the right of the leading 1 in the row above it.
• Zero rows at the bottom: Any rows consisting entirely of zeroes are at the bottom of the matrix.

Once the augmented matrix is in reduced row-echelon form, the solution to the linear system can be read directly from the matrix. The columns corresponding to the variables will have a leading 1, and the values in the rightmost column will be the solutions for those variables. If a column does not have a leading 1, the corresponding variable is a free variable, meaning it can take on any value, and the system has infinitely many solutions.

Setting up the Augmented Matrix: The First Key Step

Okay, so now that we've refreshed our understanding of Gauss-Jordan elimination, let's circle back to the original question. The first, and arguably most crucial, step in applying Gauss-Jordan elimination is setting up the augmented matrix. This matrix is a compact and efficient way to represent the linear system, making it easier to manipulate using elementary row operations. Think of it as translating the equations from words into a matrix form that our algorithm can understand.

The augmented matrix is essentially a combination of the coefficient matrix and the constant terms from the linear system. The coefficient matrix is formed by the coefficients of the variables in each equation, while the constant terms are the numbers on the right-hand side of the equations. We then glue these two matrices together, typically with a vertical line separating them, to create the augmented matrix.

To create the augmented matrix, we need to make sure that the linear system is written in a standard form. This means that all the variables should be on the left-hand side of the equations, and the constant terms should be on the right-hand side. Also, the variables should be in the same order in each equation. This is super important because the columns of the augmented matrix will correspond to specific variables, and if they're not in the correct order, we'll end up with a mess!

Let's break down how to construct the augmented matrix step-by-step, using the system provided in the prompt as an example.

Building the Augmented Matrix: A Step-by-Step Guide

Let's consider the given linear system:

$$\begin{array}{l} 3 w-x=2 y+z-4 \\ 9 x-y+z=10 \\ 4 w+3 y-z=7 \\ 12 x+17=2 y-z+6 \end{array}$$

Our mission is to transform this system into its augmented matrix representation. Let's tackle this step by step:

Step 1: Standardize the Equations

The very first thing we need to do is rewrite each equation in the standard form: where all the variables are on the left side and the constant terms are on the right. This involves moving terms around using basic algebraic manipulations. Remember, we want the variables in the same order in each equation (let's use w, x, y, z).

Let's take each equation one by one:

• Equation 1: 3w - x = 2y + z - 4
  Rearrange it to: 3w - x - 2y - z = -4
• Equation 2: 9x - y + z = 10.
  This one is already in good shape!
• Equation 3: 4w + 3y - z = 7.
  Also looking good.
• Equation 4: 12x + 17 = 2y - z + 6 Rearrange it to: 12x - 2y + z = -11 (Remember to subtract 17 from both sides!)

So, now our system looks like this:

$$\begin{array}{l} 3 w-x-2 y-z=-4 \\ 9 x-y+z=10 \\ 4 w+3 y-z=7 \\ 12 x-2 y+z=-11 \end{array}$$

Step 2: Extract the Coefficients and Constants

Now, we're ready to pluck out the coefficients of the variables and the constant terms. Remember, each column in our matrix will correspond to a variable (w, x, y, z), and the last column will hold the constants.

• Equation 1: 3w - x - 2y - z = -4
  Gives us the row: 3 -1 -2 -1 | -4
• Equation 2: 9x - y + z = 10
  Important! Notice there's no w term. We treat that as a coefficient of 0.
  So we get: 0 9 -1 1 | 10
• Equation 3: 4w + 3y - z = 7
  Missing an x term, so it's 0: 4 0 3 -1 | 7
• Equation 4: 12x - 2y + z = -11
  Missing a w term, so it's 0: 0 12 -2 1 | -11

Step 3: Assemble the Augmented Matrix

Finally, we put it all together! We arrange the rows we extracted into a matrix, separating the coefficient part from the constant terms with a vertical line. This line is just a visual aid to remind us that the last column represents the constants.

So, the augmented matrix for our system is:

$$\left[ \begin{array}{cccc|c} 3 & -1 & -2 & -1 & -4 \\ 0 & 9 & -1 & 1 & 10 \\ 4 & 0 & 3 & -1 & 7 \\ 0 & 12 & -2 & 1 & -11 \end{array} \right]$$

That's it! We've successfully transformed our linear system into its augmented matrix representation. This matrix is now ready to be fed into the Gauss-Jordan elimination machine!

Common Mistakes and Pro Tips

Before we wrap up, let's touch on a few common pitfalls and some handy tips to keep in mind when constructing augmented matrices:

• Forgetting to Standardize: This is a big one! If you don't get the equations into the standard form (variables on the left, constants on the right, variables in the same order), your matrix will be incorrect, and the Gauss-Jordan elimination will lead to the wrong solution. Double-check this step!
• Missing Coefficients: Don't forget to include 0s as placeholders for missing variables. If an equation doesn't have a particular variable, its coefficient is 0, and that 0 needs to be in the matrix.
• Sign Errors: Be super careful with signs when rearranging equations and extracting coefficients. A simple sign error can throw off the entire process.
• Organization is Key: Keep your work neat and organized. Label your rows and columns if it helps. A messy matrix is a recipe for mistakes.
• Pro Tip: Always double-check your augmented matrix against the original system of equations. Make sure each entry in the matrix corresponds to the correct coefficient or constant.

Why is the Augmented Matrix So Important?

You might be wondering,