Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.2: Direct Methods For Solving Linear Systems
Problem 4CEXP
Related questions
Question
![### Using Cramer's Rule to Solve a System of Linear Equations
#### Given System of Equations:
\[4x_1 - x_2 + x_3 = -12\]
\[2x_1 + 2x_2 + 3x_3 = 9\]
\[5x_1 - 2x_2 + 6x_3 = 8\]
Cramer's Rule is a mathematical algorithm used to solve systems of linear equations with as many equations as unknowns, using determinants.
#### Steps to Apply Cramer's Rule:
1. **Define the Coefficient Matrix \(A\):**
\[
A = \begin{pmatrix}
4 & -1 & 1 \\
2 & 2 & 3 \\
5 & -2 & 6
\end{pmatrix}
\]
2. **Define the Determinant of Matrix \(A\) (\(\det(A)\)):**
\[
\det(A) = ?
\]
3. **Define the Matrices \(A_1\), \(A_2\), and \(A_3\):**
- **Matrix \(A_1\):**
Replace the first column of \(A\) with the constants from the right-hand side of the equations.
\[
A_1 = \begin{pmatrix}
-12 & -1 & 1 \\
9 & 2 & 3 \\
8 & -2 & 6
\end{pmatrix}
\]
- **Matrix \(A_2\):**
Replace the second column of \(A\) with the constants.
\[
A_2 = \begin{pmatrix}
4 & -12 & 1 \\
2 & 9 & 3 \\
5 & 8 & 6
\end{pmatrix}
\]
- **Matrix \(A_3\):**
Replace the third column of \(A\) with the constants.
\[
A_3 = \begin{pmatrix}
4 & -1 & -12 \\
2 & 2 & 9 \\
5 & -2 & 8
\end{pmatrix}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18cc342f-1f53-4805-905e-f1f8d9083158%2F1e06a2f8-dd1c-4792-9edf-aade38ae280d%2Fp7ke2wv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Using Cramer's Rule to Solve a System of Linear Equations
#### Given System of Equations:
\[4x_1 - x_2 + x_3 = -12\]
\[2x_1 + 2x_2 + 3x_3 = 9\]
\[5x_1 - 2x_2 + 6x_3 = 8\]
Cramer's Rule is a mathematical algorithm used to solve systems of linear equations with as many equations as unknowns, using determinants.
#### Steps to Apply Cramer's Rule:
1. **Define the Coefficient Matrix \(A\):**
\[
A = \begin{pmatrix}
4 & -1 & 1 \\
2 & 2 & 3 \\
5 & -2 & 6
\end{pmatrix}
\]
2. **Define the Determinant of Matrix \(A\) (\(\det(A)\)):**
\[
\det(A) = ?
\]
3. **Define the Matrices \(A_1\), \(A_2\), and \(A_3\):**
- **Matrix \(A_1\):**
Replace the first column of \(A\) with the constants from the right-hand side of the equations.
\[
A_1 = \begin{pmatrix}
-12 & -1 & 1 \\
9 & 2 & 3 \\
8 & -2 & 6
\end{pmatrix}
\]
- **Matrix \(A_2\):**
Replace the second column of \(A\) with the constants.
\[
A_2 = \begin{pmatrix}
4 & -12 & 1 \\
2 & 9 & 3 \\
5 & 8 & 6
\end{pmatrix}
\]
- **Matrix \(A_3\):**
Replace the third column of \(A\) with the constants.
\[
A_3 = \begin{pmatrix}
4 & -1 & -12 \\
2 & 2 & 9 \\
5 & -2 & 8
\end{pmatrix}
\]
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