2. Use the equation 2 3 X1 0 +2 -3 + x3 2 + x4 -9 -7 3 0 2 for the following problems. (a) Write a system of equations that is equivalent to the given vector equation. (b) Use the definition of Ax to write the vector equation as a matrix equation. =
2. Use the equation 2 3 X1 0 +2 -3 + x3 2 + x4 -9 -7 3 0 2 for the following problems. (a) Write a system of equations that is equivalent to the given vector equation. (b) Use the definition of Ax to write the vector equation as a matrix equation. =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**2. Using Vector and Matrix Equations to Solve Linear Systems**
Given the vector equation:
\[ x_1 \begin{bmatrix} 0 \\ 0 \\ -1 \end{bmatrix} + x_2 \begin{bmatrix} 2 \\ -3 \\ 3 \end{bmatrix} + x_3 \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix} + x_4 \begin{bmatrix} 3 \\ -9 \\ 2 \end{bmatrix} = \begin{bmatrix} 3 \\ -7 \\ -3 \end{bmatrix} \]
Answer the following problems:
**(a) Write a system of equations that is equivalent to the given vector equation.**
To write a system of equations from the given vector equation, equate each component of the vectors:
\[
\begin{aligned}
1. & \quad -x_1 + 2x_2 + x_3 + 3x_4 = 3 \\
2. & \quad 0 = -7 \\
3. & \quad 0 = -3 \\
4. & \quad 2x_2 + 2x_3 - 9x_4 = -7 \\
5. & \quad 3x_2 - 7x_3 + 2x_4 = 7 \\
\end{aligned}
\]
**(b) Use the definition of \(Ax\) to write the vector equation as a matrix equation.**
The given vector equation translates into a matrix equation of the form \(Ax = b\).
The matrix \(A\), the vector \(x\), and the result vector \(b\) are:
\[ A = \begin{bmatrix} 0 & 2 & 1 & 3 \\ 0 & -3 & 2 & -9 \\ -1 & 3 & 0 & 2 \end{bmatrix} \]
\[ x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \]
\[ b = \begin{bmatrix} 3 \\ -7 \\ -3 \end{bmatrix} \]
So, the matrix equation is:
\[ \begin{bmatrix} 0 & 2 & 1 &](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d07e113-70c7-44ea-a54a-4dad627b44c8%2F0d433159-bd31-48c8-93a7-f1978dbaabd5%2Fy0tlhwi_processed.png&w=3840&q=75)
Transcribed Image Text:**2. Using Vector and Matrix Equations to Solve Linear Systems**
Given the vector equation:
\[ x_1 \begin{bmatrix} 0 \\ 0 \\ -1 \end{bmatrix} + x_2 \begin{bmatrix} 2 \\ -3 \\ 3 \end{bmatrix} + x_3 \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix} + x_4 \begin{bmatrix} 3 \\ -9 \\ 2 \end{bmatrix} = \begin{bmatrix} 3 \\ -7 \\ -3 \end{bmatrix} \]
Answer the following problems:
**(a) Write a system of equations that is equivalent to the given vector equation.**
To write a system of equations from the given vector equation, equate each component of the vectors:
\[
\begin{aligned}
1. & \quad -x_1 + 2x_2 + x_3 + 3x_4 = 3 \\
2. & \quad 0 = -7 \\
3. & \quad 0 = -3 \\
4. & \quad 2x_2 + 2x_3 - 9x_4 = -7 \\
5. & \quad 3x_2 - 7x_3 + 2x_4 = 7 \\
\end{aligned}
\]
**(b) Use the definition of \(Ax\) to write the vector equation as a matrix equation.**
The given vector equation translates into a matrix equation of the form \(Ax = b\).
The matrix \(A\), the vector \(x\), and the result vector \(b\) are:
\[ A = \begin{bmatrix} 0 & 2 & 1 & 3 \\ 0 & -3 & 2 & -9 \\ -1 & 3 & 0 & 2 \end{bmatrix} \]
\[ x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \]
\[ b = \begin{bmatrix} 3 \\ -7 \\ -3 \end{bmatrix} \]
So, the matrix equation is:
\[ \begin{bmatrix} 0 & 2 & 1 &
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