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 constant matrix.
The augmented matrix solver simplifies the solution of systems of linear equations, combining the coefficient matrix and constants from the equations into a single matrix.
What is an Augmented Matrix?
Augmented matrix is a method that is used for matrices which 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 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).
Structure of an Augmented Matrix:
Consider a system of linear equations of x valued in an nth order. Then, the augmented matrix calculator gives the result of values. Where 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 simplifies the calculation process of linear equations and provide an easy way to solve the value of variables, present in the equation.
By following the 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 to get the solution of 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 of system will again changed to system of linear equations to find value of given variables.
You can see the below example in which the above process has been used for the calculation of system of linear equations and understand how the matrix augmented calculator works.
Example of Augmented Matrix:
Lets understand the augmented matrix method with the help of an example to know how the augmented matrix calculator with steps works.
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, the value of variables (x, y, z) = (3, 0, -1) for 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 only. To understand how, here are some steps.
- Choose the size of 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 of augmented matrices calculator to get the exact solution of linear equation system.
- Click the “Calculate” button to get the solution of linear system variable values.
- If you want to check the augmented matrix solution calculator then use the load example option.
- Click the “Recalculate” button to get the solution of examples of augmented matrix question.
Outcome of Augmented Matrix Solver:
The augmented matrix calculator with steps provides you the solution of augment matrix as per your input values. It includes as:
Result Box:
When you click on the result button you get the solution in variable values form.
Steps Box:
Click on the steps option so that you get the solution of linear equation questions in a steps.
Advantage of Augmented Matrices Calculator:
The matrix augmented calculator has multiple advantages whenever you use it to solve the augment matrix problems. You just give it an input value and get a solution.
- The augmented matrix solver is a trustworthy tool that always provides you accurate solutions of system of linear equation questions.
- It is an efficient tool that evaluates augment matrix problems with solutions in seconds.
- Augmented matrix solution calculator is a learning tool that helps children to learn the concept of argument matrix online easily.
- It is a handy tool that can solve different types of equations and find the value of variables quickly.
- It is a free tool that allows you to calculate augnment matrix problems.
- It is an easy-to-use tool, anyone or even a beginner can easily use it to get the solution of augmented matrix problems.
- Augmented matrix calculator with steps can be used in all devices desktop, mobile, or laptop through the internet.
