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 *2=2 b. What are x, and x₂ when k, = 4 and k₂ = 9? 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 *2=2 b. What are x, and x₂ when k, = 4 and k₂ = 9? 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 a System of Equations Using Matrix Inverses
Consider the following system of equations:
\[
\begin{align*}
x_1 - 2x_2 & = k_1 \\
-x_1 + x_2 & = k_2
\end{align*}
\]
We aim to write this system as a matrix equation and solve it using matrix inverses.
### Matrix Formulation
The system can be represented in matrix form as:
\[
\begin{bmatrix}
1 & -2 \\
-1 & 1
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
k_1 \\
k_2
\end{bmatrix}
\]
### Solving the System
To find \( x_1 \) and \( x_2 \), we can use the inverse of the coefficient matrix. Let’s denote the coefficient matrix by **A** and the column matrix of variables by **X** and the right-hand side constants by **B**:
\[
A = \begin{bmatrix}
1 & -2 \\
-1 & 1
\end{bmatrix}, \quad
X = \begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}, \quad
B = \begin{bmatrix}
k_1 \\
k_2
\end{bmatrix}
\]
The solution is given by:
\[
X = A^{-1}B
\]
### Examples
#### a. Solving for \( k_1 = -4 \) and \( k_2 = 2 \)
\[
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
0 \\
2
\end{bmatrix}
\]
Thus, \( x_1 = 0 \) and \( x_2 = 2 \).
#### b. Solving for \( k_1 = 4 \) and \( k_2 = 9 \)
\[
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
\boxed{} \\
\boxed{}
\end{bmatrix}
\]
Thus, we need to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F49e1171a-c632-491d-85cd-df729265f508%2Fb0cfd662-1809-4841-91a9-2c00772dc671%2Fe053xml_processed.png&w=3840&q=75)
Transcribed Image Text:## Solving a System of Equations Using Matrix Inverses
Consider the following system of equations:
\[
\begin{align*}
x_1 - 2x_2 & = k_1 \\
-x_1 + x_2 & = k_2
\end{align*}
\]
We aim to write this system as a matrix equation and solve it using matrix inverses.
### Matrix Formulation
The system can be represented in matrix form as:
\[
\begin{bmatrix}
1 & -2 \\
-1 & 1
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
k_1 \\
k_2
\end{bmatrix}
\]
### Solving the System
To find \( x_1 \) and \( x_2 \), we can use the inverse of the coefficient matrix. Let’s denote the coefficient matrix by **A** and the column matrix of variables by **X** and the right-hand side constants by **B**:
\[
A = \begin{bmatrix}
1 & -2 \\
-1 & 1
\end{bmatrix}, \quad
X = \begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}, \quad
B = \begin{bmatrix}
k_1 \\
k_2
\end{bmatrix}
\]
The solution is given by:
\[
X = A^{-1}B
\]
### Examples
#### a. Solving for \( k_1 = -4 \) and \( k_2 = 2 \)
\[
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
0 \\
2
\end{bmatrix}
\]
Thus, \( x_1 = 0 \) and \( x_2 = 2 \).
#### b. Solving for \( k_1 = 4 \) and \( k_2 = 9 \)
\[
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
\boxed{} \\
\boxed{}
\end{bmatrix}
\]
Thus, we need to
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