E1 - Augmented Matrices


Example 1

E1 - Augmented Matrices (ver. 1)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrr|r} 1 & 1 & -6 & 8 \\ 0 & 1 & -5 & 6 \\ -2 & 1 & 4 & -5 \\ 0 & -1 & 2 & -3 \end{array}\right]\]

Answer.

\begin{align*} x_{1} + x_{2} - 6 \, x_{3} &= 8 \\ x_{2} - 5 \, x_{3} &= 6 \\ -2 \, x_{1} + x_{2} + 4 \, x_{3} &= -5 \\ -x_{2} + 2 \, x_{3} &= -3 \\ \end{align*}

Example 2

E1 - Augmented Matrices (ver. 2)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrr|r} 0 & 0 & 0 & 0 \\ 2 & 7 & 8 & -7 \\ -2 & -6 & -6 & 6 \\ -1 & -3 & -3 & 3 \end{array}\right]\]

Answer.

\begin{align*} 0 &= 0 \\ 2 \, x + 7 \, y + 8 \, z &= -7 \\ -2 \, x - 6 \, y - 6 \, z &= 6 \\ -x - 3 \, y - 3 \, z &= 3 \\ \end{align*}

Example 3

E1 - Augmented Matrices (ver. 3)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrrr|r} 1 & 4 & 0 & 1 & 5 \\ 0 & 0 & 1 & -3 & -6 \\ 0 & 0 & 1 & -2 & -4 \end{array}\right]\]

Answer.

\begin{align*} x_{1} + 4 \, x_{2} + x_{4} &= 5 \\ x_{3} - 3 \, x_{4} &= -6 \\ x_{3} - 2 \, x_{4} &= -4 \\ \end{align*}

Example 4

E1 - Augmented Matrices (ver. 4)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} -x_{2} &= 2 \\ x_{1} + x_{3} &= -2 \\ 2 \, x_{1} + 2 \, x_{3} &= -4 \\ -3 \, x_{1} + 7 \, x_{2} - 3 \, x_{3} &= -8 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 0 & -1 & 0 & 2 \\ 1 & 0 & 1 & -2 \\ 2 & 0 & 2 & -4 \\ -3 & 7 & -3 & -8 \end{array}\right]\]


Example 5

E1 - Augmented Matrices (ver. 5)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrrr|r} 1 & 1 & -3 & -3 & -7 \\ 0 & 1 & -3 & -4 & -5 \\ 1 & 1 & -2 & -3 & -5 \end{array}\right]\]

Answer.

\begin{align*} x_{1} + x_{2} - 3 \, x_{3} - 3 \, x_{4} &= -7 \\ x_{2} - 3 \, x_{3} - 4 \, x_{4} &= -5 \\ x_{1} + x_{2} - 2 \, x_{3} - 3 \, x_{4} &= -5 \\ \end{align*}

Example 6

E1 - Augmented Matrices (ver. 6)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrr|r} 1 & 1 & -3 & -2 \\ -5 & -4 & 0 & -5 \\ 0 & 0 & 1 & 1 \\ 3 & 4 & -6 & -3 \end{array}\right]\]

Answer.

\begin{align*} x + y - 3 \, z &= -2 \\ -5 \, x - 4 \, y &= -5 \\ z &= 1 \\ 3 \, x + 4 \, y - 6 \, z &= -3 \\ \end{align*}

Example 7

E1 - Augmented Matrices (ver. 7)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrr|r} 1 & -1 & -3 & -5 \\ 2 & -1 & -2 & -3 \\ -2 & 1 & 3 & 4 \\ 1 & 0 & -2 & -1 \end{array}\right]\]

Answer.

\begin{align*} x - y - 3 \, z &= -5 \\ 2 \, x - y - 2 \, z &= -3 \\ -2 \, x + y + 3 \, z &= 4 \\ x - 2 \, z &= -1 \\ \end{align*}

Example 8

E1 - Augmented Matrices (ver. 8)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} x_{1} + x_{2} - 3 \, x_{3} &= -2 \\ -3 \, x_{1} - 2 \, x_{2} + 5 \, x_{3} &= -1 \\ -2 \, x_{1} - 2 \, x_{2} + 7 \, x_{3} &= 6 \\ -3 \, x_{1} - x_{2} + 4 \, x_{3} &= -2 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 1 & 1 & -3 & -2 \\ -3 & -2 & 5 & -1 \\ -2 & -2 & 7 & 6 \\ -3 & -1 & 4 & -2 \end{array}\right]\]


Example 9

E1 - Augmented Matrices (ver. 9)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} x - 5 \, y &= 1 \\ z &= 1 \\ x - 5 \, y + 5 \, z &= 6 \\ x - 5 \, y + 3 \, z &= 4 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 1 & -5 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & -5 & 5 & 6 \\ 1 & -5 & 3 & 4 \end{array}\right]\]


Example 10

E1 - Augmented Matrices (ver. 10)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} -7 \, x - 4 \, y + 5 \, z &= 6 \\ 3 \, x + y &= -4 \\ 2 \, x - y + 5 \, z &= -6 \\ 4 \, x + 2 \, y - 2 \, z &= -4 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} -7 & -4 & 5 & 6 \\ 3 & 1 & 0 & -4 \\ 2 & -1 & 5 & -6 \\ 4 & 2 & -2 & -4 \end{array}\right]\]


Example 11

E1 - Augmented Matrices (ver. 11)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrrr|r} -1 & 5 & 5 & 8 & -6 \\ 0 & 0 & 0 & 1 & -1 \\ 1 & -5 & -5 & -3 & 1 \end{array}\right]\]

Answer.

\begin{align*} -x + 5 \, y + 5 \, z + 8 \, {w} &= -6 \\ {w} &= -1 \\ x - 5 \, y - 5 \, z - 3 \, {w} &= 1 \\ \end{align*}

Example 12

E1 - Augmented Matrices (ver. 12)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrrr|r} 3 & -8 & -7 & -5 & -8 \\ 2 & -5 & -4 & -4 & -5 \\ 0 & 3 & 7 & -8 & 3 \end{array}\right]\]

Answer.

\begin{align*} 3 \, x_{1} - 8 \, x_{2} - 7 \, x_{3} - 5 \, x_{4} &= -8 \\ 2 \, x_{1} - 5 \, x_{2} - 4 \, x_{3} - 4 \, x_{4} &= -5 \\ 3 \, x_{2} + 7 \, x_{3} - 8 \, x_{4} &= 3 \\ \end{align*}

Example 13

E1 - Augmented Matrices (ver. 13)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} z - {w} &= -4 \\ y + 2 \, z - 2 \, {w} &= -7 \\ -x - z + 3 \, {w} &= 3 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 0 & 0 & 1 & -1 & -4 \\ 0 & 1 & 2 & -2 & -7 \\ -1 & 0 & -1 & 3 & 3 \end{array}\right]\]


Example 14

E1 - Augmented Matrices (ver. 14)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrrr|r} -1 & 4 & 7 & 7 & 4 \\ -1 & 3 & 6 & 5 & 3 \\ -2 & 2 & 8 & 2 & 2 \end{array}\right]\]

Answer.

\begin{align*} -x_{1} + 4 \, x_{2} + 7 \, x_{3} + 7 \, x_{4} &= 4 \\ -x_{1} + 3 \, x_{2} + 6 \, x_{3} + 5 \, x_{4} &= 3 \\ -2 \, x_{1} + 2 \, x_{2} + 8 \, x_{3} + 2 \, x_{4} &= 2 \\ \end{align*}

Example 15

E1 - Augmented Matrices (ver. 15)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrr|r} 1 & 1 & -3 & -3 \\ 2 & -1 & -8 & 6 \\ 0 & -1 & -1 & 4 \\ 1 & 0 & -5 & 1 \end{array}\right]\]

Answer.

\begin{align*} x + y - 3 \, z &= -3 \\ 2 \, x - y - 8 \, z &= 6 \\ -y - z &= 4 \\ x - 5 \, z &= 1 \\ \end{align*}

Example 16

E1 - Augmented Matrices (ver. 16)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} -x + y + 3 \, z - 3 \, {w} &= -7 \\ -x + 2 \, {w} &= -2 \\ -y - 2 \, z + 3 \, {w} &= 4 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} -1 & 1 & 3 & -3 & -7 \\ -1 & 0 & 0 & 2 & -2 \\ 0 & -1 & -2 & 3 & 4 \end{array}\right]\]


Example 17

E1 - Augmented Matrices (ver. 17)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrrr|r} -1 & 0 & 2 & -1 & -1 \\ 1 & -1 & -3 & 4 & -2 \\ -2 & 2 & 6 & -8 & 4 \end{array}\right]\]

Answer.

\begin{align*} -x + 2 \, z - {w} &= -1 \\ x - y - 3 \, z + 4 \, {w} &= -2 \\ -2 \, x + 2 \, y + 6 \, z - 8 \, {w} &= 4 \\ \end{align*}

Example 18

E1 - Augmented Matrices (ver. 18)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} -x_{1} - 6 \, x_{2} + 5 \, x_{3} &= 8 \\ -x_{1} - 3 \, x_{2} + 2 \, x_{3} &= 5 \\ -x_{2} + x_{3} &= 1 \\ x_{1} + 5 \, x_{2} - 4 \, x_{3} &= -7 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} -1 & -6 & 5 & 8 \\ -1 & -3 & 2 & 5 \\ 0 & -1 & 1 & 1 \\ 1 & 5 & -4 & -7 \end{array}\right]\]


Example 19

E1 - Augmented Matrices (ver. 19)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrrr|r} -1 & 2 & 1 & 2 & -3 \\ 2 & -3 & -3 & -8 & 4 \\ 1 & -2 & -2 & -5 & 3 \end{array}\right]\]

Answer.

\begin{align*} -x_{1} + 2 \, x_{2} + x_{3} + 2 \, x_{4} &= -3 \\ 2 \, x_{1} - 3 \, x_{2} - 3 \, x_{3} - 8 \, x_{4} &= 4 \\ x_{1} - 2 \, x_{2} - 2 \, x_{3} - 5 \, x_{4} &= 3 \\ \end{align*}

Example 20

E1 - Augmented Matrices (ver. 20)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} x - 2 \, z + 6 \, {w} &= -5 \\ x + y - z + 3 \, {w} &= 0 \\ -x + 3 \, z - 8 \, {w} &= 8 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & 0 & -2 & 6 & -5 \\ 1 & 1 & -1 & 3 & 0 \\ -1 & 0 & 3 & -8 & 8 \end{array}\right]\]


Example 21

E1 - Augmented Matrices (ver. 21)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrrr|r} 1 & 3 & -1 & -6 & -2 \\ 0 & 1 & -1 & -1 & 0 \\ -1 & -1 & -1 & 4 & 2 \end{array}\right]\]

Answer.

\begin{align*} x + 3 \, y - z - 6 \, {w} &= -2 \\ y - z - {w} &= 0 \\ -x - y - z + 4 \, {w} &= 2 \\ \end{align*}

Example 22

E1 - Augmented Matrices (ver. 22)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} -x - y - 2 \, z - 3 \, {w} &= 2 \\ -x - 2 \, y - 3 \, z - 5 \, {w} &= 3 \\ 2 \, x + 3 \, y + 5 \, z + 8 \, {w} &= -5 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} -1 & -1 & -2 & -3 & 2 \\ -1 & -2 & -3 & -5 & 3 \\ 2 & 3 & 5 & 8 & -5 \end{array}\right]\]


Example 23

E1 - Augmented Matrices (ver. 23)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrrr|r} 1 & 2 & 1 & -2 & -3 \\ -1 & -2 & 2 & -7 & -6 \\ 0 & 0 & 2 & -6 & -6 \end{array}\right]\]

Answer.

\begin{align*} x + 2 \, y + z - 2 \, {w} &= -3 \\ -x - 2 \, y + 2 \, z - 7 \, {w} &= -6 \\ 2 \, z - 6 \, {w} &= -6 \\ \end{align*}

Example 24

E1 - Augmented Matrices (ver. 24)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} 2 \, x_{1} + x_{2} + 3 \, x_{3} &= -1 \\ x_{1} + x_{2} + 3 \, x_{3} - x_{4} &= 2 \\ 3 \, x_{1} + x_{2} + 4 \, x_{3} + x_{4} &= -3 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 2 & 1 & 3 & 0 & -1 \\ 1 & 1 & 3 & -1 & 2 \\ 3 & 1 & 4 & 1 & -3 \end{array}\right]\]


Example 25

E1 - Augmented Matrices (ver. 25)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} x_{1} - x_{3} - 3 \, x_{4} &= 2 \\ 2 \, x_{1} + x_{2} - 5 \, x_{4} &= -3 \\ -x_{1} - x_{2} + 2 \, x_{4} &= 2 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & 0 & -1 & -3 & 2 \\ 2 & 1 & 0 & -5 & -3 \\ -1 & -1 & 0 & 2 & 2 \end{array}\right]\]


Example 26

E1 - Augmented Matrices (ver. 26)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrr|r} 0 & 1 & -3 & -5 \\ 0 & 1 & 0 & 1 \\ 1 & -1 & 0 & 0 \\ -1 & 6 & -2 & 1 \end{array}\right]\]

Answer.

\begin{align*} y - 3 \, z &= -5 \\ y &= 1 \\ x - y &= 0 \\ -x + 6 \, y - 2 \, z &= 1 \\ \end{align*}

Example 27

E1 - Augmented Matrices (ver. 27)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrrr|r} 1 & -1 & -4 & 6 & 2 \\ 0 & 1 & 3 & -5 & -1 \\ 1 & -1 & -3 & 4 & 2 \end{array}\right]\]

Answer.

\begin{align*} x - y - 4 \, z + 6 \, {w} &= 2 \\ y + 3 \, z - 5 \, {w} &= -1 \\ x - y - 3 \, z + 4 \, {w} &= 2 \\ \end{align*}

Example 28

E1 - Augmented Matrices (ver. 28)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} x_{1} - x_{2} - 2 \, x_{3} - 4 \, x_{4} &= -3 \\ x_{2} - x_{3} &= -4 \\ -x_{1} + 4 \, x_{2} + 5 \, x_{4} &= -7 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & -1 & -2 & -4 & -3 \\ 0 & 1 & -1 & 0 & -4 \\ -1 & 4 & 0 & 5 & -7 \end{array}\right]\]


Example 29

E1 - Augmented Matrices (ver. 29)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} x - 5 \, y - 2 \, z &= 7 \\ y + z &= -3 \\ y + 2 \, z &= -6 \\ -2 \, y &= 0 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 1 & -5 & -2 & 7 \\ 0 & 1 & 1 & -3 \\ 0 & 1 & 2 & -6 \\ 0 & -2 & 0 & 0 \end{array}\right]\]


Example 30

E1 - Augmented Matrices (ver. 30)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} x - y - 4 \, z - 2 \, {w} &= 7 \\ y + 5 \, z + 4 \, {w} &= -6 \\ z + {w} &= -1 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & -1 & -4 & -2 & 7 \\ 0 & 1 & 5 & 4 & -6 \\ 0 & 0 & 1 & 1 & -1 \end{array}\right]\]


Example 31

E1 - Augmented Matrices (ver. 31)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} x_{1} + 4 \, x_{2} + 8 \, x_{3} - 3 \, x_{4} &= 6 \\ -x_{1} - 5 \, x_{3} - x_{4} &= -8 \\ -3 \, x_{2} - 2 \, x_{3} + 3 \, x_{4} &= 2 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & 4 & 8 & -3 & 6 \\ -1 & 0 & -5 & -1 & -8 \\ 0 & -3 & -2 & 3 & 2 \end{array}\right]\]


Example 32

E1 - Augmented Matrices (ver. 32)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} 4 \, x_{1} + x_{2} + 7 \, x_{3} &= -5 \\ -x_{1} - 3 \, x_{3} &= 0 \\ x_{1} + 4 \, x_{3} &= 1 \\ x_{1} + 3 \, x_{3} &= 0 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 4 & 1 & 7 & -5 \\ -1 & 0 & -3 & 0 \\ 1 & 0 & 4 & 1 \\ 1 & 0 & 3 & 0 \end{array}\right]\]


Example 33

E1 - Augmented Matrices (ver. 33)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrrr|r} 1 & 1 & 1 & 2 & -4 \\ 0 & 1 & 2 & 1 & -2 \\ -1 & 2 & 5 & 1 & -2 \end{array}\right]\]

Answer.

\begin{align*} x + y + z + 2 \, {w} &= -4 \\ y + 2 \, z + {w} &= -2 \\ -x + 2 \, y + 5 \, z + {w} &= -2 \\ \end{align*}

Example 34

E1 - Augmented Matrices (ver. 34)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} x_{1} - 5 \, x_{2} - 2 \, x_{3} &= -8 \\ -x_{1} + 6 \, x_{2} + 3 \, x_{3} &= 8 \\ -x_{1} + 5 \, x_{2} + 3 \, x_{3} &= 5 \\ -5 \, x_{2} - 6 \, x_{3} &= 3 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 1 & -5 & -2 & -8 \\ -1 & 6 & 3 & 8 \\ -1 & 5 & 3 & 5 \\ 0 & -5 & -6 & 3 \end{array}\right]\]


Example 35

E1 - Augmented Matrices (ver. 35)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} x_{1} - 4 \, x_{2} + 4 \, x_{3} + x_{4} &= -8 \\ x_{1} - 3 \, x_{2} + 4 \, x_{3} + 3 \, x_{4} &= -6 \\ x_{3} + 2 \, x_{4} &= 0 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & -4 & 4 & 1 & -8 \\ 1 & -3 & 4 & 3 & -6 \\ 0 & 0 & 1 & 2 & 0 \end{array}\right]\]


Example 36

E1 - Augmented Matrices (ver. 36)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrr|r} 0 & 0 & -2 & 0 \\ 2 & -6 & -7 & -2 \\ 1 & -3 & -2 & -1 \\ 1 & -3 & -2 & -1 \end{array}\right]\]

Answer.

\begin{align*} -2 \, z &= 0 \\ 2 \, x - 6 \, y - 7 \, z &= -2 \\ x - 3 \, y - 2 \, z &= -1 \\ x - 3 \, y - 2 \, z &= -1 \\ \end{align*}

Example 37

E1 - Augmented Matrices (ver. 37)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrr|r} 5 & 3 & -4 & 1 \\ -2 & -1 & 1 & 1 \\ -4 & -3 & 6 & -7 \\ 1 & 1 & -2 & 3 \end{array}\right]\]

Answer.

\begin{align*} 5 \, x + 3 \, y - 4 \, z &= 1 \\ -2 \, x - y + z &= 1 \\ -4 \, x - 3 \, y + 6 \, z &= -7 \\ x + y - 2 \, z &= 3 \\ \end{align*}

Example 38

E1 - Augmented Matrices (ver. 38)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrrr|r} 1 & 1 & 1 & -2 & -1 \\ 0 & 1 & 2 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

Answer.

\begin{align*} x + y + z - 2 \, {w} &= -1 \\ y + 2 \, z - {w} &= 1 \\ 0 &= 0 \\ \end{align*}

Example 39

E1 - Augmented Matrices (ver. 39)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrr|r} 3 & -8 & -6 & -4 \\ -2 & 1 & 4 & 7 \\ -2 & 3 & 4 & 5 \\ -1 & 4 & 2 & 0 \end{array}\right]\]

Answer.

\begin{align*} 3 \, x - 8 \, y - 6 \, z &= -4 \\ -2 \, x + y + 4 \, z &= 7 \\ -2 \, x + 3 \, y + 4 \, z &= 5 \\ -x + 4 \, y + 2 \, z &= 0 \\ \end{align*}

Example 40

E1 - Augmented Matrices (ver. 40)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} 2 \, x + 8 \, y - 2 \, {w} &= 4 \\ x + 4 \, y + 3 \, z - 4 \, {w} &= 8 \\ x + 4 \, y - 2 \, z + {w} &= -2 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 2 & 8 & 0 & -2 & 4 \\ 1 & 4 & 3 & -4 & 8 \\ 1 & 4 & -2 & 1 & -2 \end{array}\right]\]


Example 41

E1 - Augmented Matrices (ver. 41)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} -x - z + {w} &= 0 \\ x - 3 \, {w} &= 1 \\ 2 \, x - y - 8 \, {w} &= 2 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} -1 & 0 & -1 & 1 & 0 \\ 1 & 0 & 0 & -3 & 1 \\ 2 & -1 & 0 & -8 & 2 \end{array}\right]\]


Example 42

E1 - Augmented Matrices (ver. 42)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} x_{1} - x_{3} + x_{4} &= -3 \\ x_{2} + 3 \, x_{3} - 7 \, x_{4} &= 4 \\ -x_{2} - 2 \, x_{3} + 5 \, x_{4} &= -3 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & 0 & -1 & 1 & -3 \\ 0 & 1 & 3 & -7 & 4 \\ 0 & -1 & -2 & 5 & -3 \end{array}\right]\]


Example 43

E1 - Augmented Matrices (ver. 43)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrrr|r} 1 & 4 & -3 & 7 & -7 \\ 0 & 1 & -2 & 0 & 0 \\ -1 & -2 & 0 & -6 & 6 \end{array}\right]\]

Answer.

\begin{align*} x_{1} + 4 \, x_{2} - 3 \, x_{3} + 7 \, x_{4} &= -7 \\ x_{2} - 2 \, x_{3} &= 0 \\ -x_{1} - 2 \, x_{2} - 6 \, x_{4} &= 6 \\ \end{align*}

Example 44

E1 - Augmented Matrices (ver. 44)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} x_{1} - x_{2} + 3 \, x_{3} + 3 \, x_{4} &= 2 \\ -2 \, x_{1} + x_{2} - 5 \, x_{3} - 3 \, x_{4} &= -4 \\ x_{2} - x_{3} - 3 \, x_{4} &= 0 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & -1 & 3 & 3 & 2 \\ -2 & 1 & -5 & -3 & -4 \\ 0 & 1 & -1 & -3 & 0 \end{array}\right]\]


Example 45

E1 - Augmented Matrices (ver. 45)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrr|r} 1 & 2 & -4 & 8 \\ -1 & -1 & 2 & -3 \\ 0 & 2 & -3 & 7 \\ -1 & 0 & 2 & -4 \end{array}\right]\]

Answer.

\begin{align*} x + 2 \, y - 4 \, z &= 8 \\ -x - y + 2 \, z &= -3 \\ 2 \, y - 3 \, z &= 7 \\ -x + 2 \, z &= -4 \\ \end{align*}

Example 46

E1 - Augmented Matrices (ver. 46)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrrr|r} 0 & -2 & 6 & -3 & -7 \\ 0 & 1 & -3 & 2 & 5 \\ -1 & -1 & 0 & -2 & -7 \end{array}\right]\]

Answer.

\begin{align*} -2 \, y + 6 \, z - 3 \, {w} &= -7 \\ y - 3 \, z + 2 \, {w} &= 5 \\ -x - y - 2 \, {w} &= -7 \\ \end{align*}

Example 47

E1 - Augmented Matrices (ver. 47)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} -2 \, x_{1} - 2 \, x_{2} + 8 \, x_{3} &= -2 \\ x_{2} - 2 \, x_{3} &= -1 \\ x_{1} - 2 \, x_{3} &= 2 \\ 2 \, x_{1} - 3 \, x_{2} + 2 \, x_{3} &= 7 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} -2 & -2 & 8 & -2 \\ 0 & 1 & -2 & -1 \\ 1 & 0 & -2 & 2 \\ 2 & -3 & 2 & 7 \end{array}\right]\]


Example 48

E1 - Augmented Matrices (ver. 48)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} -x_{1} - 3 \, x_{2} - 3 \, x_{3} - 3 \, x_{4} &= 8 \\ x_{2} - x_{3} + 5 \, x_{4} &= -2 \\ x_{1} + 3 \, x_{2} + 2 \, x_{3} + 5 \, x_{4} &= -8 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} -1 & -3 & -3 & -3 & 8 \\ 0 & 1 & -1 & 5 & -2 \\ 1 & 3 & 2 & 5 & -8 \end{array}\right]\]


Example 49

E1 - Augmented Matrices (ver. 49)

Give a linear system that is represented by the following matrix.

\[\left[\begin{array}{rrrr|r} 0 & -1 & -1 & -2 & 3 \\ 2 & -1 & -3 & -8 & -5 \\ -1 & 0 & 1 & 3 & 4 \end{array}\right]\]

Answer.

\begin{align*} -x_{2} - x_{3} - 2 \, x_{4} &= 3 \\ 2 \, x_{1} - x_{2} - 3 \, x_{3} - 8 \, x_{4} &= -5 \\ -x_{1} + x_{3} + 3 \, x_{4} &= 4 \\ \end{align*}

Example 50

E1 - Augmented Matrices (ver. 50)

Give an augmented matrix that represents the following system of linear equations.

\begin{align*} 4 \, x - 8 \, y + 4 \, z &= 0 \\ 2 \, x - 3 \, y + z &= 1 \\ -x + 7 \, y - 6 \, z &= 5 \\ -x + 2 \, y - z &= 0 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 4 & -8 & 4 & 0 \\ 2 & -3 & 1 & 1 \\ -1 & 7 & -6 & 5 \\ -1 & 2 & -1 & 0 \end{array}\right]\]