An augmented matrix is a matrix that combines the coefficients of a system of linear equations with the constants on the right-hand side. It provides a compact way to represent and solve systems of linear equations using row operations.
Solved Examples
Example 1: Find the augmented matrix of the system of equations.
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.
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.
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.
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.
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.
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.
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