The free augmented matrix calculator is a specialized tool designed to help you solve systems of linear equations using augmented matrices.
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What Is an Augmented Matrix?
But what exactly is an augmented matrix? An augmented matrix is created by combining two matrices into a single matrix. It's a technique used to solve systems of linear equations. The number of rows in an augmented matrix always matches the number of variables in the linear equation.
To grasp this concept better, let's explore it with the help of three linear equations:
1. a1x + b1y + c1z = d1
2. a2x + b2y + c2z = d2
3. a3x + b3y + c3z = d3
Here, we have three matrices involved:
- Matrix of Coefficients – A = [Math Processing Error]
- Matrix of Constant Terms – B = [Math Processing Error]
- Matrix of Variables – C = [Math Processing Error]
Solving the Augmented Matrix: A Step-by-Step Guide
Now, how do we go about solving the augmented matrix? Let's use an example:
Example of Augmented Matrix:
Imagine we have the following system of linear equations:
1. 3x + 5y = 10
2. 7x + 9y = 15
To make things easier, you can use an augmented matrix calculator, preferably one with a 2x3 setup. However, we'll also walk through the manual calculations:
Step 1: Divide the first row by 3: R0 = R0/3
[Math Processing Error]
Step 2: Multiply the first row by 7 and subtract it from the second row: R1 = R1 - 7R0
[Math Processing Error]
Step 3: Multiply the second row by 3/-8: R1 = 3/-8 R1
[Math Processing Error]
Step 4: Multiply the second row by 5/3 and subtract it from the first row: R0 = R0 - 5/3R1
The resulting augmented matrix in reduced echelon form looks like this:
[1 0 -15/8]
[0 1 25/8]
Properties of Augmented Matrices:
- The number of columns in an augmented matrix is determined by the variables in the linear equations and the constant terms.
- The number of rows in the augmented matrix matches the number of systems of equations.
- You can swap rows in the augmented matrix.
- You can multiply or divide the elements of a specific row by a constant.
- You can add or subtract one row from another.
- You can apply the multiple of one matrix row to another matrix row.
To work with our augmented matrix calculator, follow these steps:
Input:
1. Specify the matrix's order.
2. Enter the matrix elements.
3. Click the calculate button.
Output:
1. Get detailed steps for solving the augmented matrix.
2. Find the solution to your linear equation.
