Write the given system of equations as a matrix equation and solve by using inverses. X₁ - 2x₂ K₁ -X₁ + x2 k₂ a. What are x₁ and x₂ when k₁= -4 and k₂ =2? x₁ = 0 x2 = 2 b. What are x, and x₂ when k₁ = 4 and k₂ = 9? x₁ = -22 x₂ = -13 c. What are x₁ and x₂ when k₁ = 5 and k₂ = 3? X₁ x2 = =
Write the given system of equations as a matrix equation and solve by using inverses. X₁ - 2x₂ K₁ -X₁ + x2 k₂ a. What are x₁ and x₂ when k₁= -4 and k₂ =2? x₁ = 0 x2 = 2 b. What are x, and x₂ when k₁ = 4 and k₂ = 9? x₁ = -22 x₂ = -13 c. What are x₁ and x₂ when k₁ = 5 and k₂ = 3? X₁ x2 = =
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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![### Solving Systems of Equations Using Matrix Inverses
In this exercise, we will solve a system of linear equations by expressing it as a matrix equation and using the matrix inverse.
Given the system of equations:
\[ x_1 - 2x_2 = k_1 \]
\[ -x_1 + x_2 = k_2 \]
We can express this system as a matrix equation \( A \mathbf{x} = \mathbf{k} \), where:
\[ A = \begin{pmatrix} 1 & -2 \\ -1 & 1 \end{pmatrix} \]
\[ \mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \]
\[ \mathbf{k} = \begin{pmatrix} k_1 \\ k_2 \end{pmatrix} \]
To solve for \( \mathbf{x} \), we use the inverse of matrix \( A \), if it exists:
\[ \mathbf{x} = A^{-1} \mathbf{k} \]
Let's solve for \( x_1 \) and \( x_2 \) for the given values of \( k_1 \) and \( k_2 \).
#### a. When \( k_1 = -4 \) and \( k_2 = 2 \):
\[ \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = A^{-1} \begin{pmatrix} -4 \\ 2 \end{pmatrix} \]
Calculated values:
\[ x_1 = 0 \]
\[ x_2 = 2 \]
#### b. When \( k_1 = 4 \) and \( k_2 = 9 \):
\[ \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = A^{-1} \begin{pmatrix} 4 \\ 9 \end{pmatrix} \]
Calculated values:
\[ x_1 = -22 \]
\[ x_2 = -13 \]
#### c. When \( k_1 = 5 \) and \( k_2 = 3 \):
\[ \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = A^{-1} \begin{pmatrix} 5 \\ 3 \end{pmatrix} \]
Calculated values](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F49e1171a-c632-491d-85cd-df729265f508%2F662e6ca7-b21c-4edf-adbf-2fc0bf39ba69%2F5187wko_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Systems of Equations Using Matrix Inverses
In this exercise, we will solve a system of linear equations by expressing it as a matrix equation and using the matrix inverse.
Given the system of equations:
\[ x_1 - 2x_2 = k_1 \]
\[ -x_1 + x_2 = k_2 \]
We can express this system as a matrix equation \( A \mathbf{x} = \mathbf{k} \), where:
\[ A = \begin{pmatrix} 1 & -2 \\ -1 & 1 \end{pmatrix} \]
\[ \mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \]
\[ \mathbf{k} = \begin{pmatrix} k_1 \\ k_2 \end{pmatrix} \]
To solve for \( \mathbf{x} \), we use the inverse of matrix \( A \), if it exists:
\[ \mathbf{x} = A^{-1} \mathbf{k} \]
Let's solve for \( x_1 \) and \( x_2 \) for the given values of \( k_1 \) and \( k_2 \).
#### a. When \( k_1 = -4 \) and \( k_2 = 2 \):
\[ \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = A^{-1} \begin{pmatrix} -4 \\ 2 \end{pmatrix} \]
Calculated values:
\[ x_1 = 0 \]
\[ x_2 = 2 \]
#### b. When \( k_1 = 4 \) and \( k_2 = 9 \):
\[ \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = A^{-1} \begin{pmatrix} 4 \\ 9 \end{pmatrix} \]
Calculated values:
\[ x_1 = -22 \]
\[ x_2 = -13 \]
#### c. When \( k_1 = 5 \) and \( k_2 = 3 \):
\[ \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = A^{-1} \begin{pmatrix} 5 \\ 3 \end{pmatrix} \]
Calculated values
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