How to Form an Augmented Matrix

An augmented matrix is a powerful tool in linear algebra that helps solve systems of linear equations. It combines the coefficients and constants of the equations into a single matrix, making it easier to manipulate and solve.

Step-by-Step Guide

1. Write Down the System of Equations

Consider the system of linear equations:

$begin{cases}
2x + 3y = 5 \
4x – y = 1 \
end{cases}$

2. Identify the Coefficients and Constants

For each equation, identify the coefficients of the variables and the constants. In our example, the coefficients and constants are:

• First equation: Coefficients are 2 and 3, constant is 5.
• Second equation: Coefficients are 4 and -1, constant is 1.

3. Form the Coefficient Matrix

Create a matrix using the coefficients of the variables:

$begin{bmatrix}
2 & 3 \
4 & -1 \
end{bmatrix}$

4. Form the Constant Matrix

Create a matrix using the constants from the equations:

$begin{bmatrix}
5 \
1 \
end{bmatrix}$

5. Combine into Augmented Matrix

Combine the coefficient matrix and the constant matrix to form the augmented matrix. This is done by placing the constant matrix as an additional column to the right of the coefficient matrix:

$begin{bmatrix}
2 & 3 & | & 5 \
4 & -1 & | & 1 \
end{bmatrix}$

The vertical bar (|) is often used to separate the coefficients from the constants, but it’s not always necessary.

Why Use Augmented Matrices?

Augmented matrices simplify the process of solving systems of linear equations. They allow us to use methods like Gaussian elimination or Gauss-Jordan elimination to find solutions systematically.

Example

Let’s solve the system using the augmented matrix:

1. Original Augmented Matrix:

$begin{bmatrix}
2 & 3 & | & 5 \
4 & -1 & | & 1 \
end{bmatrix}$

2. Row Operations:
   • Divide the first row by 2:

   $begin{bmatrix}
   1 & 1.5 & | & 2.5 \
   4 & -1 & | & 1 \
   end{bmatrix}$

   • Subtract 4 times the first row from the second row:

   $begin{bmatrix}
   1 & 1.5 & | & 2.5 \
   0 & -7 & | & -9 \
   end{bmatrix}$

   • Divide the second row by -7:

   $begin{bmatrix}
   1 & 1.5 & | & 2.5 \
   0 & 1 & | & 1.2857 \
   end{bmatrix}$

   • Subtract 1.5 times the second row from the first row:

   $begin{bmatrix}
   1 & 0 & | & 0.5714 \
   0 & 1 & | & 1.2857 \
   end{bmatrix}$

3. Solution:
   • From the final matrix, we get:

   $x = 0.5714, y = 1.2857$

Conclusion

Understanding how to form and use an augmented matrix is crucial for solving systems of linear equations efficiently. It provides a structured way to handle and manipulate the equations, leading to a clearer path to the solution.

Citations

1. 1. Khan Academy – Matrices
2. 2. Math is Fun – Matrices and Determinants
3. 3. Purplemath – Augmented Matrices