Augmented Matrix (Practice Problems)

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.

\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