Advanced Engineering Mathematics
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ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Solving Systems of Linear Equations Using Matrix Capabilities and Cramer's Rule**
In this tutorial, you will learn how to solve the following system of linear equations using a software program or a graphing utility with matrix capabilities and Cramer's Rule.
The system of equations given is:
1. \( 3x_1 - 2x_2 + 9x_3 + 4x_4 = 41 \)
2. \( -x_1 - 9x_3 - 6x_4 = -23 \)
3. \( 3x_3 + x_4 = 11 \)
4. \( 2x_1 + 2x_2 + 8x_4 = -34 \)
To solve this system, follow these steps:
1. **Set Up the Coefficient Matrix:**
The coefficient matrix \( A \) for the system is:
\[
A = \begin{bmatrix}
3 & -2 & 9 & 4 \\
-1 & 0 & -9 & -6 \\
0 & 0 & 3 & 1 \\
2 & 2 & 0 & 8
\end{bmatrix}
\]
2. **Set Up the Constants Column Matrix:**
The constants column matrix \( B \) is:
\[
B = \begin{bmatrix}
41 \\
-23 \\
11 \\
-34
\end{bmatrix}
\]
3. **Apply Cramer's Rule:**
According to Cramer's Rule, to find each variable \( x_i \) in the system, you replace the i-th column of the coefficient matrix \( A \) with the constant matrix \( B \) and calculate the determinant. The solution values \( x_1, x_2, x_3, x_4 \) are obtained by:
\[
x_1 = \frac{\text{det}(A_1)}{\text{det}(A)}, \quad x_2 = \frac{\text{det}(A_2)}{\text{det}(A)}, \quad x_3 = \frac{\text{det}(A_3)}{\text{det}(A)}, \quad x_4 = \frac{\text{det}(](https://content.bartleby.com/qna-images/question/3e1fa912-5e0c-4d6e-bcb6-bc3942e62084/b0b1ee29-fd1d-473e-a1bd-2074022ad04a/yo6vpb_processed.jpeg)
Transcribed Image Text:**Solving Systems of Linear Equations Using Matrix Capabilities and Cramer's Rule**
In this tutorial, you will learn how to solve the following system of linear equations using a software program or a graphing utility with matrix capabilities and Cramer's Rule.
The system of equations given is:
1. \( 3x_1 - 2x_2 + 9x_3 + 4x_4 = 41 \)
2. \( -x_1 - 9x_3 - 6x_4 = -23 \)
3. \( 3x_3 + x_4 = 11 \)
4. \( 2x_1 + 2x_2 + 8x_4 = -34 \)
To solve this system, follow these steps:
1. **Set Up the Coefficient Matrix:**
The coefficient matrix \( A \) for the system is:
\[
A = \begin{bmatrix}
3 & -2 & 9 & 4 \\
-1 & 0 & -9 & -6 \\
0 & 0 & 3 & 1 \\
2 & 2 & 0 & 8
\end{bmatrix}
\]
2. **Set Up the Constants Column Matrix:**
The constants column matrix \( B \) is:
\[
B = \begin{bmatrix}
41 \\
-23 \\
11 \\
-34
\end{bmatrix}
\]
3. **Apply Cramer's Rule:**
According to Cramer's Rule, to find each variable \( x_i \) in the system, you replace the i-th column of the coefficient matrix \( A \) with the constant matrix \( B \) and calculate the determinant. The solution values \( x_1, x_2, x_3, x_4 \) are obtained by:
\[
x_1 = \frac{\text{det}(A_1)}{\text{det}(A)}, \quad x_2 = \frac{\text{det}(A_2)}{\text{det}(A)}, \quad x_3 = \frac{\text{det}(A_3)}{\text{det}(A)}, \quad x_4 = \frac{\text{det}(
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