Introduction to Augmented Matrix Calculator:
The augmented matrix calculator with steps is a powerful tool that is used to make a new matrix after taking the elements from the linear equation and the constant matrix.
Our tool simplifies the solution of systems of linear equations after combining the coefficient matrix and constants from the equations into a single matrix.
The augmented matrix solver is a beneficial tool because it helps you to provide the solution of complex calculations of a system of linear equations quickly and easily.
What is an Augmented Matrix?
Augmented matrix is a method that is used for matrices in which it combines the coefficient of a system of linear equations with the constants and makes them a new matrix.
The resulting augmented matrix again changes into the original system of equations in which we can easily find the value of variables. This method provides an easy way to solve the system of equations with other methods (like Gaussian elimination) for solution.
Structure of an Augmented Matrix:
Consider a system of linear equations of x valued in an nth order. When the augmented matrix calculator gives the result then the values b1,..., bn represent the constant terms.
$$ a_{11}x_1 + a_{12}x_2 + … + a_{1n} x_n \;=\; b_1 $$
$$ a_{21}x_1 + a_{22}x_2 + … + a_{2n} x_n \;=\; b_2 $$
The coefficient and cofactor of the matrix is represented in matrix A and constant in matrix B.
$$ A \;=\; \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{matrix}, b \;=\; \begin{matrix}b_1 \\ b_2 \\ \vdots \\ b_m \\ \end{matrix} $$
When the elements from matrix A and matrix b are combined the augmented matrix becomes,
\begin{array}{rrrr|r} a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\ a_{21} & a_{22} & \cdots & a_{2n} & b_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \\ \end{array}
How to Calculate the Augmented Matrix?
The augmented matrices calculator can simplify the calculation process of linear equations and provide an easy way to solve the value of variables that are present in the given equation.
By following these steps of augmented matrix solution calculator, you will know how to find the augmented matrix for any given system of linear equations.
Step 1:
Identify the coefficients of the variables and the constants in the system of linear equation.
Step 2:
To form the Augmented Matrix combine all the coefficients and constants term in a new matrix for the solution of the given system.
Step 3:
After making an augmented matrix use row operations (multiply, add/subtract, and divide rows) to transform the matrix into echelon form.
Step 4:
Once the matrix obtains echelon form, the solutions to the system can be again changed into a system of linear equations from the matrix to find the value of given variables.
You can see the below example in which the above process has been used for the calculation of a system of linear equations and understand how the matrix augmented calculator works.
Practical Example of an Augmented Matrix:
Lets understand the augmented matrix method used by the augmented matrix calculator with steps with the help of a solved example that will clear all your doubts related to the calculations.
For Example:
Use the augmented matrix to transform the following system of linear equations into triangular form.
$$ \left\{ \begin{array}{c} 3x\; - \; y\; +\; z \;=\; 8 \\ x \;+ 2y\; - \; z \;=\; 4 \\ 2x \;+ 3y \; 4z \;=\; 10 \\ \end{array} \right. $$
Solution:
$$ \left\{ \begin{array}{c} 3x\; - \; y\; +\; z \;=\; 8 \\ x \;+ 2y\; - \; z \;=\; 4 \\ 2x \;+ 3y \; 4z \;=\; 10 \\ \end{array} \right. $$
Change the given system of equations into an augmented matrix (into a new matrix) where the coefficient of variable or constant is involved.
$$ \left[ \begin{array}{rrr|r} 3 & -1 & 1 & 8 \\ 1 & 2 & -1 & 4 \\ 2 & 3 & -4 & 10 \\ \end{array} \right] $$
To change the given matrix into echelon form use the guss elimination method,
$$ \left[ \begin{array}{rrr|r} 3 & -1 & 1 & 8 \\ 1 & 2 & -1 & 4 \\ 2 & 3 & -4 & 10 \\ \end{array} \right] \xrightarrow{Switch\; R_1\; and\; R_2} \left[ \begin{array}{rrr|r} 1 & 2 & -1 & 4 \\ 3 & -1 & 1 & 8 \\ 2 & 3 & -4 & 10 \\ \end{array} \right] $$
Replace R3 with -2R1 + R3,
$$ \left[ \begin{array}{rrr|r} 1 & 2 & -1 & 4 \\ 3 & -1 & 1 & 8 \\ 2 & 3 & -4 & 10 \\ \end{array} \right] \xrightarrow{Replace\; R_2\; with \; -3R1 + R2} \left[ \begin{array}{rrr|r} 1 & 2 & -1 & 4 \\ 0 & -7 & 4 & -4 \\ 0 & -1 & -2 & 2 \\ \end{array} \right] $$
$$ \left[ \begin{array}{rrr|r} 1 & 2 & -1 & 4 \\ 0 & -7 & 4 & -4 \\ 0 & -1 & -2 & 2 \\ \end{array} \right] \xrightarrow{Replace\; R_2\; with \; -1/7 R2} \left[ \begin{array}{rrr|r} 1 & 2 & -1 & 4 \\ 0 & 1 & -4/7 & 4/7 \\ 0 & -1 & -2 & 2 \\ \end{array} \right] $$
$$ \left[ \begin{array}{rrr|r} 1 & 2 & -1 & 4 \\ 0 & 1 & -4/7 & 4/7 \\ 0 & -1 & -2 & 2 \\ \end{array} \right] \xrightarrow{Replace\; R3\; with \; R2+R3} \left[ \begin{array}{rrr|r} 1 & 2 & -1 & 4 \\ 0 & 1 & -4/7 & 4/7 \\ 0 & 0 & -18/7 & 18/7 \\ \end{array} \right] $$
$$ \left[ \begin{array}{rrr|r} 1 & 2 & -1 & 4 \\ 0 & 1 & -4/7 & 4/7 \\ 0 & 0 & -18/7 & 18/7 \\ \end{array} \right] \xrightarrow{Replace\; R3\; with \; -7/18 R3} \left[ \begin{array}{rrr|r} 1 & 2 & -1 & 4 \\ 0 & 1 & -4/7 & 4/7 \\ 0 & 0 & 1 & -1 \\ \end{array} \right] $$
Now the matrix can achieve the echelon form so again convert them into a system of equations to find the value of its variable.
$$ x + 2y - z \;=\; 4 $$
$$ y - \frac{4}{7}z \;=\; \frac{4}{7} $$
$$ z \;=\; -1 $$
As we have z=-1, so for the value of y put the z value in equation 2 as:
$$ y \;=\; \frac{4}{7}z + \frac{4}{7} $$
Put z=-1 we get the value of y,
$$ =\; \frac{4}{7}(-1) + \frac{4}{7} \;=\; 0 $$
y=0
For the value of x, put the y or z value in equation 1 to get the x value.
$$ x \;=\; -2y + z + 4 $$
Put z=-1 ,y=0
$$ =\;-2(0) + (-1) + 4 \;=\; 3 $$
x=3
So after combing the value of variable (x,y,z)=(3,0,-1) of the given linear equation system.
How to Use the Augmented Matrix Calculator?
The augmented matrix solver has a user-friendly layout that helps you to find the value from the given equation instantly.
You just need to put your problem in this augmented matrices calculator only by following some of our guidelines that keep you away from any trouble. These steps are:
- Choose the size of the matrix as per your system of equation.
- Enter the coefficient of the system of equations in the input field.
- Review your given input value before clicking the calculate button to get the exact solution of the linear equation system with an augmented matrix.
- Click the “Calculate” button for the solution of variable values from linear system problems.
- If you want to check the augmented matrix solution calculator then use the load example to get an idea about its evaluation process.
- Click the “Recalculate” button for the solution of more examples of the augmented matrix question.
Outcome of Augmented Matrix Solver:
The augmented matrix calculator with steps provides you the solution for finding the values from the augment matrix as per your input values when you click on the calculate button. It may include as:
In the Result Box,
When you click on the result button you get the solution in the form of variable values.
Steps Box
Click on the steps option so that you get the solution of linear equation questions in a step-by-step method.
Advantage of Augmented Matrices Calculator:
The matrix augmented calculator has tons of advantages whenever you use it to solve the augment matrix problems for finding the variable values in solution. You just need to add the input value and get a solution without imposing any condition.
- The augmented matrix solver is a trustworthy tool that always provides you with accurate solutions to the system of linear equation questions
- It is an efficient tool that evaluates augment matrix problems with solutions in a run of time
- Augmented matrix solution calculator is a learning tool that helps children about the concept of argument matrix very easily on online platforms without going to a teacher.
- It is a handy tool that can solve different types of equations to find the value of variables quickly without putting external effort.
- It is a free tool that allows you to use it for the calculation of augnment matrix problems.
- It is an easy-to-use tool, anyone or even a beginner can easily use it for the solution of augmented matrix problems.
- Augmented matrix calculator with steps operate through all devices desktop, mobile, or laptop on the internet to solve augumented matrix problems.
