Augmented Matrix

An augmented matrix is a matrix formed by combining the coefficient matrix of a system of linear equations with the column matrix of constants. It provides a compact way to represent and solve linear equations using matrix operations.

Example: Consider the following system of linear equations and its corresponding matrix representation:

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

Augmented Matrix of a System of Linear Equations

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,

1. 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}

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

3. 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 a_ij and variables x₁, x₂, x₃, ...,x_n

\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}

Steps to Form an Augmented Matrix

Agumented matrix is found by following the steps discussed below:

• Identify the coefficient matrix from the given system of equations.
• Extract the constant terms and form the constant matrix.
• Append the constant matrix to the coefficient matrix, separated by a vertical line, to obtain the augmented matrix.

Steps to Solve an 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.

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.
• The number of columns in an augmented matrix is one more than the number of variables in the 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 & 0 \\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]

Solved Examples

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

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

Related Articles

• Square Matrix
• Determinant of a Matrix
• Inverse of a Matrix