Augmented Matrix Calculator

Get the resultant variable value of the augmented matrix for two variable matrices with our augmented matrix calculator which enables you to get fast calculations of a system of simultaneous linear equations. 

What is an Augmented Matrix?

“The matrix is gained by adding the columns of two given metrics for performing the elementary row operations on the given matrices. This is called an augmented matrix”. 

This is useful when solving the operations for linear equations. This metric indicates that the number of rows is equal to the number of variables in the linear and the constant terms and the rows of an augmented matrix can be interchanged. 

How to Find Augmented Matrix?

Let us assume that there is “m” column in the first matrix and “n” column in the second matrix then in the augmented matrix there would be m + n columns. Look at the three linear equations to understand the concept of 

a1x + b1y + c1z = d1

a2x + b2y + c2z = d2

a3x + b3y + c3z = d3

Matrix of coefficient -A = 
$$ \begin{bmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{bmatrix} $$

Matrix of constant -B = 
$$ \begin{bmatrix}d_1\\d_2\\d_3\end{bmatrix} $$

Matrix of variable -C = 
$$ \begin{bmatrix}x\\y\\z\end{bmatrix} $$

This augmented matrix calculator 2x3 has the capability to enhance your calculations but manual calculations are also important. To understand the manual calculations look at the example below.

Example:

Assume that there is the following system of linear equations:

                         4x + 5y = 12

                         7x + 11y = 10

Solution:

$$ \begin{bmatrix}4 & 5 & 12 \\  7 & 11 & 10  \\\end{bmatrix} $$

Divide row 0 by 4: R0 = R0/4

$$ \left[  \begin{array}{cc|c}1& \frac{5}{4}&3  \\ 7&11&10  \\\end{array}\right] $$

In an augmented matrix, the vertical line is placed to indicate the series of equal signs and divide the term into two sides. 

Subtract row 1 multiplied by 7 from row R0: R1 = R1 - 7R0

$$ \left[\begin{array}{cc|c}1& \frac{5}{4}&3 \\ 0& \frac{9}{4}&-11 \\\end{array}\right] $$

Multiply row 1 by 4/9: R1 = 4/9 R1

$$ \left[\begin{array}{cc|c}1& \frac{5}{4}&3 \\0&1& \frac{-44}{9} \\\end{array}\right] $$

Subtract row 0 multiplied by 5/4 from row R1: R0 = R0 - 5/4R1

$$ \left[\begin{array}{cc|c}1&0& \frac{82}{9}\\0&1& \frac{-44}{9}\\\end{array}\right] $$

Working of This Augmented Matrix Calculator:

Rather than wasting your time, you should use the augmented matrix solution calculator so you can get the results you are looking for as quickly as possible.

Input:

• Select the order of a matrix 
• Put the elements of a matrix 
• Tap “Calculate” 

Output:

Discover the following conclusions with our augmented matrix solver:

• Representation of the Augmented Matrix
• Step-by-step calculation 

Properties of Augmented Matrix:

On the augmented matrix calculator 3x3 interface, you will find the coefficients correspondings to the linear equations. But here we indicate the properties of an augmented matrix:

• The multiples of matrix rows can be applicable to other matrix rows
• To multiply or divide the elements of a row constants can be used
• There is no restriction to the number of rows in the augmented matrix.
• There is the same number of rows as the number of equations of systems. 
• The number of columns is equal to the variables and the constants in the linear equation.

FAQs:

Why Vertical Line Is Used In An Augmented Matrix?

Vertical lines are used in the matrix to indicate the series of equal signs and it divides the term into two sides. 

When The Augmented Matrix Is Linearly Independent?

If the determinant is equal to zero then we say that it is linearly dependent
If the determinant is not equal to zero then it is linearly independent. 

References:

From the source Wikipedia: Augmented matrix, To find the inverse of a matrix, Existence, and number of solutions, Solution of a linear system.