Augmented Matrix System Of Equations Explained Step-by-Step

Hey everyone! Let's dive into the world of augmented matrices and how they can help us solve systems of equations. If you've ever felt lost trying to juggle multiple equations with multiple variables, you're in the right place. We're going to break down a specific example step-by-step, making it super clear and easy to follow.

Understanding Augmented Matrices

Before we jump into the example, let's quickly recap what an augmented matrix actually is. Simply put, an augmented matrix is a way to represent a system of linear equations in a compact, matrix form. It's like taking all the coefficients and constants from your equations and arranging them neatly into a table. This table then becomes our playground for performing row operations, which ultimately helps us find the solutions to the system.

The beauty of using augmented matrices lies in their efficiency. Instead of writing out the variables (like x, y, and z) repeatedly, we can focus solely on the numbers. This not only saves time but also reduces the chances of making errors. Plus, it sets the stage for using powerful techniques like Gaussian elimination and Gauss-Jordan elimination, which are the workhorses for solving larger systems of equations.

Constructing the Augmented Matrix

To construct an augmented matrix, you will need to follow a systematic approach, ensuring that each part of the equation is correctly represented in the matrix form. The process involves extracting coefficients and constants from the equations and arranging them in a specific manner. Here’s a detailed guide on how to do it:

1. Identify the Coefficients and Constants:

   • Start by examining each equation in your system. Identify the coefficients of the variables (e.g., x, y, z) and the constants (the numbers on the right side of the equals sign).
   • For example, in the equation 2x + 3y - z = 5, the coefficients are 2, 3, and -1 (for x, y, and z respectively), and the constant is 5.

2. Arrange the Coefficients in a Matrix:

   • Create a matrix where each row corresponds to an equation, and each column corresponds to a variable. The last column is reserved for the constants.
   • If an equation does not have a particular variable, use 0 as its coefficient in the matrix. This ensures that all variables are accounted for in the matrix structure.
   • For instance, if you have the equations:
     • x + y + z = 6
     • 2x - y + z = 1
     • x + 3y - 2z = -2
   • The coefficient matrix would look like this:

   | 1  1  1 |
   | 2 -1  1 |
   | 1  3 -2 |
   

3. Add the Constants as the Last Column:

   • Draw a vertical line (often dashed or solid) to separate the coefficient matrix from the constants. This line visually distinguishes the coefficients from the constants.
   • Place the constants from each equation in the last column, corresponding to the respective rows.
   • For the equations mentioned above, the augmented matrix would be:

   | 1  1  1 | 6 |
   | 2 -1  1 | 1 |
   | 1  3 -2 | -2|
   

4. Ensure Proper Alignment:

   • Make sure that coefficients of the same variable are aligned in the same column. This is crucial for performing row operations later.
   • Double-check that the constants are placed in the correct row, corresponding to their respective equations.

By following these steps meticulously, you can accurately construct an augmented matrix from any system of linear equations. This matrix representation is the foundation for solving systems using methods like Gaussian elimination, which we’ll discuss later. Remember, a well-constructed augmented matrix is half the battle won in solving linear systems!

Our System of Equations

Okay, let's get to the system we're tackling today:

{ x - y + 10z = 10
  y - 6z = -6
  z = 2 }

This system has three equations and three variables: x, y, and z. Our mission is to represent this system as an augmented matrix. Remember, the goal is to organize the coefficients and constants in a way that makes it easy to manipulate and solve the system.

Identifying Coefficients and Constants

The first step in transforming our system of equations into an augmented matrix is to carefully identify the coefficients of the variables (x, y, and z) and the constants on the right-hand side of each equation. This is a foundational step, and accuracy here is crucial for the rest of the process. Let's break down each equation:

1. First Equation: x - y + 10z = 10
   • The coefficient of x is 1 (since x is the same as 1x).
   • The coefficient of y is -1 (since -y is the same as -1y).
   • The coefficient of z is 10.
   • The constant on the right side is 10.
2. Second Equation: y - 6z = -6
   • Here, we need to be a bit careful. Notice that x is missing from this equation. When a variable is absent, we consider its coefficient to be 0.
   • So, the coefficient of x is 0.
   • The coefficient of y is 1.
   • The coefficient of z is -6.
   • The constant is -6.
3. Third Equation: z = 2
   • Again, we have missing variables. Both x and y are absent, so their coefficients are 0.
   • The coefficient of x is 0.
   • The coefficient of y is 0.
   • The coefficient of z is 1.
   • The constant is 2.

By meticulously identifying these coefficients and constants, we lay the groundwork for constructing the augmented matrix. This step ensures that we have all the necessary pieces in place and understand their roles in the matrix. It’s like gathering all your ingredients before you start cooking – having everything ready makes the process smoother and ensures a better outcome. Next, we'll arrange these numbers into the matrix format, bringing us one step closer to solving the system!

Building the Augmented Matrix Step-by-Step

Now that we've identified all the coefficients and constants, we're ready to assemble them into the augmented matrix. Remember, an augmented matrix is essentially a table where each row represents an equation, and each column represents the coefficients of a specific variable (x, y, z) or the constants.

Here's how we'll build it:

1. Setting up the Matrix Structure:
   • Since we have three equations, our matrix will have three rows.
   • We have three variables (x, y, z) plus the constants, so our matrix will have four columns. The last column will be separated by a vertical line to distinguish the constants from the coefficients.
2. Filling in the First Row (First Equation):
   • Our first equation is x - y + 10z = 10.
   • The coefficients are 1 (for x), -1 (for y), and 10 (for z).
   • The constant is 10.
   • So, the first row of our matrix will be: | 1 -1 10 | 10 |
3. Filling in the Second Row (Second Equation):
   • Our second equation is y - 6z = -6.
   • Remember, the coefficient of x is 0 because it's not present in the equation.
   • The coefficients are 0 (for x), 1 (for y), and -6 (for z).
   • The constant is -6.
   • The second row of our matrix will be: | 0 1 -6 | -6 |
4. Filling in the Third Row (Third Equation):
   • Our third equation is z = 2.
   • Both x and y are missing, so their coefficients are 0.
   • The coefficients are 0 (for x), 0 (for y), and 1 (for z).
   • The constant is 2.
   • The third row of our matrix will be: | 0 0 1 | 2 |

By systematically filling in each row with the coefficients and constants from our equations, we create a complete and accurate augmented matrix. This matrix now represents our system of equations in a compact and organized format, ready for the next step: solving the system using row operations.

The Final Augmented Matrix

Putting it all together, our augmented matrix for the given system of equations is:

| 1 -1 10 | 10 |
| 0  1 -6 | -6 |
| 0  0  1 |  2 |

This matrix beautifully encapsulates our original system. Each row represents an equation, and each column represents the coefficients of x, y, z, and the constants. The vertical line separates the coefficients from the constants, making it easy to see the structure.

Significance of the Augmented Matrix Structure

The structure of this augmented matrix is particularly significant because it is in row-echelon form. Notice how the leading coefficients (the first non-zero number in each row) form a sort of