Augmented Matrix
An augmented matrix is a matrix formed by combining the columns of two matrices to form a new matrix. The augmented matrix is an important tool in matrices used to solve simple linear equations. The number of rows in the augmented matrix is equal to the number of variables in the linear equation.
In this article, let us discuss the concept of an augmented matrix and its properties. We will learn how to solve augmented matrix and how it helps to solve a system of linear equations. Let us learn more about how to solve the augmented matrix, the properties of the augmented matrix, with the help of examples.
What Is An Augmented Matrix?
An augmented matrix is a means to solve simple linear equations. The coefficients and constant values of the linear equations are represented as a matrix, referred to as an augmented matrix. In simple terms, the augmented matrix is the combination of two simple matrices along the columns. If there are m columns in the first matrix and n columns in the second matrix, then there would be m + n columns in the augmented matrix.
Let us understand the concept of augmented matrix, with the help of three linear equations, represented as follows.
a_{1}x + b_{1}y + c_{1}z = d_{1}
a_{2}x + b_{2}y + c_{2}z = d_{2}
a_{3}x + b_{3}y + c_{3}z = d_{3}
The three above equations can be represented in matrix form with the coefficients as one matrix, the constant terms as another matrix, and the variables as a separate matrix.
Matrix of Coefficients  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 terms  B = \(\begin{bmatrix}d_1\\d_2\\d_3\end{bmatrix}\)
Matrix of Variables  C = \(\begin{bmatrix}x\\y\\z\end{bmatrix}\)
The augmented matrix 'M' can be represented as a matrix after combining the matrices with the coefficient terms and the constant terms.
M = [A  B]
M = \(\begin{bmatrix} a_1&b_1&c_1&d_1\\a_2&b_2&c_2&d_2\\a_3&b_3&c_3&d_3\end{bmatrix}\)
Here M is the augmented matrix and the number of rows in the augmented matrix is equal to the number of linear equations. The coefficients of the x terms are in the first column, the coefficients of the y terms are in the second column, the coefficients of the z term are in the third column, and the constant term is in the last column. The elementary row operations can be easily performed on an augmented matrix to find the solutions to the linear equations.
Augmented Matrix Meaning
An augmented matrix is a matrix that is formed by joining matrices with the same number of rows along the columns. It is used to solve a system of linear equations and to find the inverse of a matrix.
How to Solve Augmented Matrix?
The augmented matrix is solved by performing operations across its rows, and it helps to find the solution to the linear equations represented in the augmented matrix. The augmented matrix contains the coefficient values and the constant terms. Applying the Gauss Jordan Method of row transformation, the operations on rows help in transforming a part of the augmented matrix into an identity matrix. The elements remaining in the last column after the row transformations are the values of the variable of the linear equations.
Let us understand this with the notations from the equations of the line. The three equations of the lines are a_{1}x + b_{1}y + c_{1}z = d_{1}, a_{2}x + b_{2}y + c_{2}z = d_{2}, a_{3}x + b_{3}y +c_{3}z = d_{3}. Let us represent these three equations in the form of an augmented matrix.
A = \(\begin{bmatrix} a_1&b_1&c_1&d_1\\a_2&b_2&c_2&d_2\\a_3&b_3&c_3&d_3\end{bmatrix}\)
Here we can perform numerous row operations to obtain the following matrix. We apply elementary row operations to make the left side of the bar an identity matrix and the right side gives the solution to the system of equations.
A = \(\begin{bmatrix} 1&0&0&k\\0&1&0&l\\0&0&1&m\end{bmatrix}\)
Here the elements in the last row represent the values of the variables, and we have x = k, y = l, z = m, respectively.
Properties Of Augmented Matrix
The following properties help in a better understanding of an augmented matrix.
 The augmented matrix is a rectangular matrix.
 The number of columns is equal to the variables in the linear equations and the constant term.
 The number of rows is equal to the number of linear equations.
 The rows of the augmented matrix can be interchanged.
 The elements of a particular row can be multiplied or divided with a constant.
 The particular row can be added and subtracted to other rows of the matrix.
 The multiple of a row can be added to another row of the matrix.
Finding Inverse of Matrix Using Augmented Matrix
Consider a 3 × 3 matrix A = \(\begin{bmatrix} a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{bmatrix}\) and to find the inverse of the matrix A, we obtain the augmented matrix (A  I), where I is a 3 × 3 identity matrix. We apply elementary row operations on (A  I) to make the left side of the augmented matrix an identity matrix and obtain the matrix (I  A^{1}).
Important Notes on Augmented Matrix
 An augmented matrix is a matrix that is formed by joining matrices with the same number of rows along the columns.
 It is used to solve a system of linear equations and to find the inverse of a matrix.
 We can apply elementary row operations on the augmented matrix.
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Examples on Augmented Matrix

Example 1: Represent the equations 3x + 2y + z = 8, 4x  3y + 3z = 7, and x + 5y  3z = 11, as an augmented matrix.
Solution:
The three given equations are:
3x + 2y + z = 8
4x  3y + 3z = 7
x + 5y  3z = 11
Augmented Matrix = \(\begin{bmatrix} 3&2&1&8\\4&3&3&7\\1&5&3&11\end{bmatrix}\)
Thus the coefficient and constant terms of the three equations have been represented as an augmented matrix.

Example 2: Solve the equations 4x + 3y = 11, and 5x  3y = 7, using augmented matrix.
Solution:
The given linear equations are 4x + 3y = 11, and 5x  3y = 7. These equations can be represented as the following augmented matrix.
A = \(\begin{bmatrix}4&3&11\\5&3&7\end{bmatrix}\)
Here we perform the row operation:
R_{1}→ R_{1} + R_{2}
A = \(\begin{bmatrix}9&0&18\\5&3&7\end{bmatrix}\)
R_{1}→ 1/9.R_{1}
A = \(\begin{bmatrix}1&0&2\\5&3&7\end{bmatrix}\)
R_{2}→ R_{2}  5R_{1}
A = \(\begin{bmatrix}1&0&2\\0&3&3\end{bmatrix}\)
R_{2}→ (1/3).R_{2}
A = \(\begin{bmatrix}1&0&2\\0&1&1\end{bmatrix}\)
Thus the last column represents the values of the variables and we have x = 2, and y = 1.
Therefore with the help of the row operations on the augmented matrix, we could solve and find the variables of the equations.

Example 3: Find the inverse of matrix A = \(\begin{bmatrix}1&3\\5&0\end{bmatrix}\) using augmented matrix.
Solution: To find the inverse of A, we have
(A  I) = \(\begin{bmatrix}1&31&0\\5&00&1\end{bmatrix}\)
We apply row operations to obtain an identity matrix on the left side of the matrix.
R_{2} → R_{2} + 5R_{1}
\(\begin{bmatrix}1&31&0\\0&155&1\end{bmatrix}\)
R_{2} → (1/15)R_{2}
\(\begin{bmatrix}1&31&0\\0&11/3&1/15\end{bmatrix}\)
R_{1} → R_{1}  3R_{2}
(I  A^{1}) = \(\begin{bmatrix}1&00&1/5\\0&13&1/15\end{bmatrix}\)
Hence, the inverse of A is,
A^{1} = \(\begin{bmatrix}0&1/5\\3&1/15\end{bmatrix}\)
FAQs on Augmented Matrix
What Is Augmented Matrix In Algebra?
The augmented matrix is a representation of the linear equations in matrix form and is used to find the solutions of the linear equations. The linear equations ax + by = c, and px + qy = r, can be represented as an augmented matrix as A = \(\begin{bmatrix}a&b&c\\p&q&r\end{bmatrix}\). Here the coefficient of the x terms is represented in the first column, the coefficient of the y term is represented in the second column, and the constant term is represented in the last column.
How Do You Represent An Augmented Matrix?
The augmented matrix represents the coefficients of the variables in the linear equations and the constant terms of the linear equations, in a rectangular matrix format. The linear equations a_{1}x + b_{1}y + c_{1}z = d_{1}, a_{2}x + b_{2}y + c_{2}z = d_{2}, a_{3}x + b_{3}y +c_{3}z = d_{3}, can be represented as an augmented matrix as follows.
A = \(\begin{bmatrix} a_1&b_1&c_1&d_1\\a_2&b_2&c_2&d_2\\a_3&b_3&c_3&d_3\end{bmatrix}\)
The number of rows is equal to the number of linear equations, and the number of columns is equal to the number of variables and the constant term.
How To Solve An Augmented Matrix?
The augmented matrix is solved through performing rows operations, using the Guass Jordan method. The augmented matrix A = \(\begin{bmatrix} a_1&b_1&c_1&d_1\\a_2&b_2&c_2&d_2\\a_3&b_3&c_3&d_3\end{bmatrix}\) is simplified through performing numerous row operations, to obtain A = \(\begin{bmatrix} 1&0&0&k\\0&1&0&l\\0&0&1&m\end{bmatrix}\). Here the part of the augmented matrix represents the identity matrix, and the last column represents the values of the variable present in the linear equations.
What Row Operations Can Be Performed On An Augmented Matrix?
The following important row operations can be performed on an augmented matrix.
 The rows of the augmented matrix can be interchanged.
 The elements of a particular row can be multiplied or divided with a constant.
 The particular row can be added and subtracted to other rows of the matrix.
 The multiple of a row can be added to another row of the matrix.
What Is The Use Of Augmented Matrix?
The augmented matrix is useful to represent the coefficients of the variables and the constant terms of the linear equations as a matrix and to solve and find the values of the variables, through performing row operations. We can also use the augmented matrix method to find the inverse of a matrix.
What is an Augmented Matrix Method?
An augmented matrix method is a method in algebra that is used to solve a system of linear equations. Each row of the augmented matrix represents an equation of the system.
How to Find Rank of Augmented Matrix?
We can find the rank of an augmented matrix by performing elementary row operations on an augmented matrix and counting the number of rows without zeros.
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