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In Exercises 24-29, display the coefficient matrix [tex]\( A \)[/tex] and the augmented matrix [tex]\( B \)[/tex] for the given system.

24.
[tex]\[
x_1 - x_2 = -1
\][/tex]

25.
[tex]\[
\begin{aligned}
x_1 + x_2 - x_3 &= 2 \\
x_1 + x_2 &= 3 \\
2x_1 - x_3 &= 1
\end{aligned}
\][/tex]

26.
[tex]\[
\begin{aligned}
x_1 + 3x_2 - x_3 &= 1 \\
2x_1 + 5x_2 + x_3 &= 5
\end{aligned}
\][/tex]

27.
[tex]\[
\begin{aligned}
x_1 + x_2 + 2x_3 &= \\
x_1 + x_2 + x_3 &= 3 \\
3x_1 + 4x_2 - x_3 &= \\
-x_1 + x_2 + x_3 &=
\end{aligned}
\][/tex]

28.
[tex]\[
\begin{aligned}
x_1 + x_2 - 3x_3 &= -1 \\
x_1 + 2x_2 - 5x_3 &= -2 \\
-x_1 - 3x_2 + 7x_3 &= 3
\end{aligned}
\][/tex]

29.
[tex]\[
\begin{array}{r}
x_1 + x_2 + x_3 = 1 \\
2x_1 + 3x_2 + x_3 = 2 \\
x_1 - x_2 + 3x_3 = 2
\end{array}
\][/tex]



Answer :

GinnyAnswer
Sure! Let's break down the coefficient matrix [tex]\(A\)[/tex] and the augmented matrix [tex]\(B\)[/tex] for each of the given systems of equations:

### Exercise 24
The system of equations is:
[tex]\[ x_1 - x_2 = -1 \][/tex]

Here, the coefficient matrix [tex]\(A\)[/tex] and the augmented matrix [tex]\(B\)[/tex] are:

[tex]\[ A = \begin{bmatrix} 1 & -1 \end{bmatrix} \][/tex]

[tex]\[ B = \begin{bmatrix} -1 \end{bmatrix} \][/tex]

### Exercise 25
The system of equations is:
[tex]\[ x_1 + x_2 - x_3 = 2 \][/tex]
[tex]\[ x_1 + x_2 = 3 \][/tex]
[tex]\[ 2x_1 - x_3 = 1 \][/tex]

Here, the coefficient matrix [tex]\(A\)[/tex] and the augmented matrix [tex]\(B\)[/tex] are:

[tex]\[ A = \begin{bmatrix} 1 & 1 & -1 \\ 1 & 1 & 0 \\ 2 & 0 & -1 \end{bmatrix} \][/tex]

[tex]\[ B = \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} \][/tex]

### Exercise 26
The system of equations is:
[tex]\[ x_1 + 3x_2 - x_3 = 1 \][/tex]
[tex]\[ 2x_1 + 5x_2 + x_3 = 5 \][/tex]

Here, the coefficient matrix [tex]\(A\)[/tex] and the augmented matrix [tex]\(B\)[/tex] are:

[tex]\[ A = \begin{bmatrix} 1 & 3 & -1 \\ 2 & 5 & 1 \end{bmatrix} \][/tex]

[tex]\[ B = \begin{bmatrix} 1 \\ 5 \end{bmatrix} \][/tex]

### Exercise 27
The system of equations is:
[tex]\[ x_1 + x_2 + 2x_3 = \][/tex]
[tex]\[ x_1 + x_2 + x_3 = 3 \][/tex]
[tex]\[ 3x_1 + 4x_2 - x_3 = \][/tex]
[tex]\[ -x_1 + x_2 + x_3 = \][/tex]

Here, the coefficient matrix [tex]\(A\)[/tex] and the augmented matrix [tex]\(B\)[/tex] are:

[tex]\[ A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & 1 & 1 \\ 3 & 4 & -1 \\ -1 & 1 & 1 \end{bmatrix} \][/tex]

[tex]\[ B = \begin{bmatrix} \text{nan} \\ 3 \\ \text{nan} \\ \text{nan} \end{bmatrix} \][/tex]

(Note: "nan" stands for "not a number," indicating unknown values.)

### Exercise 28
The system of equations is:
[tex]\[ x_1 + x_2 - 3x_3 = -1 \][/tex]
[tex]\[ x_1 + 2x_2 - 5x_3 = -2 \][/tex]
[tex]\[ -x_1 - 3x_2 + 7x_3 = 3 \][/tex]

Here, the coefficient matrix [tex]\(A\)[/tex] and the augmented matrix [tex]\(B\)[/tex] are:

[tex]\[ A = \begin{bmatrix} 1 & 1 & -3 \\ 1 & 2 & -5 \\ -1 & -3 & 7 \end{bmatrix} \][/tex]

[tex]\[ B = \begin{bmatrix} -1 \\ -2 \\ 3 \end{bmatrix} \][/tex]

### Exercise 29
The system of equations is:
[tex]\[ x_1 + x_2 + x_3 = 1 \][/tex]
[tex]\[ 2x_1 + 3x_2 + x_3 = 2 \][/tex]
[tex]\[ x_1 - x_2 + 3x_3 = 2 \][/tex]

Here, the coefficient matrix [tex]\(A\)[/tex] and the augmented matrix [tex]\(B\)[/tex] are:

[tex]\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 1 \\ 1 & -1 & 3 \end{bmatrix} \][/tex]

[tex]\[ B = \begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix} \][/tex]

This concludes the breakdown of the coefficient matrices [tex]\(A\)[/tex] and the augmented matrices [tex]\(B\)[/tex] for the given systems of equations in Exercises 24-29.

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