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.
