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Augmented Matrix

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Augmented Matrix is a matrix that is formed when we combine the columns of two matrices and thus, form a new matrix. The new matrix so formed is called the Augmented Matrix. An Augmented Matrix is important to solve various types of problems in mathematics especially those which involve the use of equations. Augmented Matrix is used to solve simple linear equations. An Augmented Matrix has the same number of rows as there are variables in the given linear equations.

This article deals with the concept of an Augmented Matrix, its properties, examples, and others in detail. it will also help us to understand how the augmented matrix is used to solve linear equations.

What are Augmented Matrices?

To solve linear equations we have developed a method in which we combine two matrices made using those equations and thus, make it easier to solve those linear equations. This combined matrix is called an Augmented Matrices. The coefficient of the linear equations and the constant value associated with those equations are combined together to form a single matrix and this matrix is called an Augmented matrix. In Augmented Matrices, these two matrices are combined using their column values. Thus, if we have m columns in the first matrix and n columns in the second matrix, then in the augmented matrix we have (m + n) columns.

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.

The image added below a 3×3 matrix A and a 3×1 matrix B. Then the augmented matrix is [A|B] as shown in the image,

Agumented-Matrix

Augmented Matrix of a System of Linear Equation

Augmented Matrix is the combination of two matrices of the system of linear equations which contains the coefficient matrix and the constant matrix (column matrix) separated by a dotted line.

Let’s understand the same concept with an example suppose we take three different linear equations,

  • a1x + b1y + c1z = d1
  • a2x + b2y + c2z = d2
  • a3x + b3y + c3z = d3

These three linear equations are represented in matrix form as,

Coefficient Matrix(A) = \begin{bmatrix}a_{1}& b_{1}&... & c_{1} \\a_{2}& b_{2}&... & c_{2}\\a_{3}& b_{3}&... & c_{3}\end{bmatrix}

Constant Matrix(B) = \begin{bmatrix}d_{1}\\d{2}\\d{3}\end{bmatrix}

Variable Matrix(X) = \begin{bmatrix}x\\y\\z\end{bmatrix}

Now, the augmented matrix M is calculated as,

M = [A|B]

where,

  • M is Augmented Matrix
  • A is Coefficient Matrix
  • B is Constant Matrix

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}

For General System of Linear Equations with coefficient aij and variables x1, x2, x3, …,xn

\left\{ \begin{array}{c} 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 \\ \hspace{0.5cm} .\hspace{0.1cm}.\hspace{0.1cm}\hspace{4cm} .\hspace{0.1cm}.\hspace{0.1cm} \\\hspace{0.5cm} .\hspace{0.1cm}.\hspace{0.1cm}\hspace{4cm} .\hspace{0.1cm}.\hspace{0.1cm}\\a_{n1}x_1+a_{n2}x_2 +.....+a_{nn}x_n=b_n\end{array} \right.

Augmented Matrix is,

 \begin{bmatrix}    a_{11}& a_{12}&... & a_{1n}| & b_1 \\a_{21}& a_{22}&... & a_{2n}| & b_2 \\    ...& ...& ...&...\hspace{0.2cm}| &... \\    a_{n1}& a_{n2}&... & a_{nn}| & b_n  \\   \end{bmatrix}

How to Find Augmented Matrix?

Agumented matrix is found by following the steps discussed below:

Step 1: Find the coefficient matrix from a given system of equations.

Step 2: Find the constant matrix from the RHS of equations.

Step 3: Combine both the matrix separated by a dotted line to obtain an augmented matrix.

Thus, following these steps, Augmented Amtrix is obtained.

How to Solve Augmented Matrix?

The solution to the system of the linear equation is easily found by simplifying the augmented matrix and transforming the same into an identity matrix by following the Gauss-Jordan Method of matrix transformation, or by simply using Row Operation or Column Operation on the augmented matrix. Using the row transformation we change the first part of the augmented matrix into the identity matrix and then values in the last column are the solution to the given linear equations.

Suppose we are given the system linear equation as

  • a1x + b1y + c1z = d1
  • a2x + b2y + c2z = d2
  • a3x + b3y + c3z = d3

Now we know that the augmented matrix(A) for the same is formed as

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}

Performing elementary operation and solving it then making the first matrix as identity matrix results in the change in the last column of the augmented matrix. Which in turn gives the result to the given system of linear equations.

The simplified matrix is written as,

A = \begin{bmatrix}1& 0 & 0| & p \\0& 1& 0| & q\\0& 0& 1| & r\end{bmatrix}

Thus, the values p, q, and r in the last column give the required answer to the system of linear equations.

  • x = p
  • y = q
  • z = r

Properties Of Augmented Matrix

Augmented matrix has various properties and some of the important properties of the augmented matrix are mentioned below:

  • Augmented matrix is always a rectangular matrix.
  • Columns in the augmented matrix are equal to the number of variables in the given system of linear equations.
  • We can interchange the rows in the Augmented matrix without actually changing the value of the augmented matrix.
  • Elementry operations are easily applied to any row and column of the augmented matrix.
  • We can multiply any row with a constant value without actually changing the value of the augmented matrix

Inverse of Matrix Using Augmented Matrix

We can also find the inverse of any matrix by using the augmented matrix concept. Suppose we have 3 × 3 matrix A such that,

A = \begin{bmatrix}a_{1}& b_{1}& c_{1} \\a_{2}& b_{2} & c_{2}\\a_{3}& b_{3} & c_{3}\end{bmatrix}

To find the inverse of the matrix we write the augmented matrix as

P = [A|I]

P = \begin{bmatrix}a_{1}& b_{1}& c_{1}| & 1 & 0 & 1 \\a_{2}& b_{2} & c_{2}| & 0 & 1 & 0\\a_{3}& b_{3} & c_{3}| & 0 & 0 & 1\end{bmatrix}

Now by using elementary operation, we change the matrix A into the Identity matrix, and the identity matrix associated with it changes into the inverse matrix as,

P = [I|A-1]

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Solved Examples on Augmented Matrix

Example 1: Find the augmented matrix of the system of equations,

\left\{ \begin{array}{c} -x-4y-9z=-7 \\ -3x-4y-5z=-4 \\ -2x-3y-6z=-3 \end{array} \right.

Solution:

Coefficient Matrix: \begin{bmatrix}    -1 & -4 & -9 \\    -3& -4 & -5 \\    -2 & -3 & -6 \\ \end{bmatrix}

Constant Matrix: \begin{bmatrix}    -7 \\    -4 \\    -3\\ \end{bmatrix}\\

Required Augmented Matrix:

\begin{bmatrix}  -1 & -4 & -9| &-7\\    -3 & -4 & -5| &-4 \\    -2 & -3 & -6| & -3  \\   \end{bmatrix}

Example 2: Find the augmented matrix of the system of equations,

\left\{ \begin{array}{c} 2x-14y+9z=47 \\ -3x+24y=-52 \\ 2x-13z=23 \end{array} \right.

Solution:

Coefficient Matrix: \begin{bmatrix} 2 & \hspace{-0.3cm}-14 & 9 \\    \hspace{-0.3cm} -3 & 24 & 0 \\    2 & 0 & \hspace{-0.3cm}-13 \\ \end{bmatrix}

Constant Matrix: \begin{bmatrix}    47 \\    \hspace{-0.3cm}-52 \\    23\\ \end{bmatrix}\\

Required Augmented Matrix: 

 \begin{bmatrix}    2 & -14 & 9| &47\\    \hspace{-0.3cm}-3 & \hspace{0.3cm}24 & 0| &\hspace{-0.4cm}-52 \\    2 & \hspace{0.3cm}0 & \hspace{-0.4cm}-13| & 23  \\   \end{bmatrix}

Example 3: Find the augmented matrix of the system of equations,

\left\{ \begin{array}{c} 9x-13y=25 \\ 20x+4y=-38 \end{array} \right.

Solution:

Coefficient Matrix: \begin{bmatrix}    9 & -13 \\    20&\hspace{0.3cm} 4  \\     \end{bmatrix}

Constant Matrix: \begin{bmatrix}   \hspace{0.3cm} 25 \\    -38 \\    \end{bmatrix}\\

Required Augmented Matrix:

\begin{bmatrix}    9 & -13 | &\hspace{0.3cm}25\\    20 &\hspace{0.4cm} 4\hspace{0.05cm} | &-38\\  \end{bmatrix}

Example 4: Find the augmented matrix of the system of equations,

\left\{ \begin{array}{c} 2a-12b-29c=-11 \\ -3a+21b+10d=28 \\ 26b-16c+8d=36 \\ a+15b-18c+5d=-14\end{array} \right.

Solution:

Coefficient Matrix:  \begin{bmatrix}    2 & \hspace{-0.3cm}-12& -29& 0 \\    \hspace{-0.3cm}-3& 21 & 0 & 10\\    0 & 26 & -16& 8\\1 & 15 & -18 & 5\\ \end{bmatrix}

Constant Matrix: \begin{bmatrix}    -11 \\    \hspace{0.3cm}28 \\    \hspace{0.3cm}36\\   -14\\ \end{bmatrix}\\

Required Augmented Matrix: 

\begin{bmatrix}    2 & \hspace{-0.3cm}-12 & -29 & 0| &-11\\    \hspace{-0.3cm}-3 & 21 & \hspace{0.3cm}0 & \hspace{-0.2cm}10\hspace{0.015cm}| & \hspace{0.3cm}28 \\    0 & 26 & -16 & 8| & \hspace{0.3cm}36  \\   1 & 15 & -18 & 5| & -14  \end{bmatrix}

Example 5: Find the augmented matrix of the system of equations,

\left\{ \begin{array}{c} -10.6x+3.1y-z=-7.8 \\ -3.2x-4.8y+1.6z=-17.2 \\ -2x-3.7y-6.6z=-8.9 \end{array} \right.

Solution:

Coefficient Matrix: \begin{bmatrix}    -10.6 & 3.1 & -1 \\    -3.2& -4.8 & 1.6 \\    -2 & -3.7 & -6.6 \\ \end{bmatrix}

Constant Matrix:  \begin{bmatrix}    -7.8 \\    -17.2 \\    -8.9\\ \end{bmatrix}\\

Required Augmented Matrix:

\begin{bmatrix}    -10.6 & 3.1 & -1| &-7.8\\    -3.2 & -4.8 & 1.6| &-17.2 \\    -2 & -3.7 & -6.6| & -8.9 \\   \end{bmatrix}

Example 6: Find the augmented matrix of the system of equations,

 \left\{ \begin{array}{c} -10x+3y=-7 \\ -3x-4y=-17 \end{array} \right.

Solution:

Coefficient Matrix: \begin{bmatrix}    -10 & 3 \\    -3& -4 \\     \end{bmatrix}

Constant Matrix:  \begin{bmatrix}    -7 \\    -17 \end{bmatrix}\\

Required Augmented Matrix: \begin{bmatrix}    -10 & 3| &-7\\    -3 & -4| &-17 \\       \end{bmatrix}

Example 7: Find the augmented matrix of the system of equations

 \left\{ \begin{array}{c} 10x+9y-z= 8 \\ 2x-8y+6z=-2 \\ -7x-3y-6z=-9 \end{array} \right.

Solution:

Coefficient Matrix: \begin{bmatrix}    \hspace{0.3cm}10&\hspace{0.3cm} 9 & -1 \\    \hspace{0.3cm}2& -8 & \hspace{0.3cm}6 \\    -7 & -3 & -6 \\ \end{bmatrix}

Constant Matrix: \begin{bmatrix}    \hspace{0.25cm}8 \\    -2 \\    -9\\ \end{bmatrix}\\

Required Augmented matrix: \begin{bmatrix}   \hspace{0.3cm} 10 & \hspace{0.3cm}9 & -1| &\hspace{0.3cm}8\\    \hspace{0.3cm}2 & -8 & \hspace{0.3cm}6| &-2 \\    -7 & -3 & -6| & \hspace{0.3cm}9 \\   \end{bmatrix}

Practice Questions on Augmented Matrix

Q1: Find the Augmented Matrix for 2x + 3y = 2 and 3x – y = 1

Q2: Find the Augmented Matrix for 2x + y = 7 and 4x – 3y + 1 = 0

Q3: Find the Augmented Matrix for 2x + 3y = 0, 3x + 4y = 5 and x + y = 1

Q4: Find the Augmented Matrix for 2x + 3y + 1 = 0 and (7 – 4x)/3 = y

FAQs on Augmented Matrix

1. What is Augmented Matrix?

A matrix formed by joining two matrices is called an augmented matrix. It is used to find the solution to the linear equations.

2. How to Represent an Augmented Matrix?

An Augumented Matrix for system of equations a1x + b1y + c1z = d1, a2x + b2y + c2z = d2 and a3x + b3y + c3z = d3 is given as M = [A|B] = \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}

3. How to Solve the Augmented Matrix?

To solve the Augmented matrix we use the elementary operation, i.e. using elementary row operation and elementary column operation results in solving the augmented matrix.

4. What is the Use of Augmented Matrix?

Augmented matrix used to find various things such as,

  • It is used to find the Inverse of the Matrix
  • It is used to solve the system of linear equations
  • It is used to find the rank of the matrix, etc

5. How to find the Rank of the Augmented Matrix?

To find the rank of the matrix we simplify the augmented matrix using the elementary operations.



Last Updated : 12 Jan, 2024
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