Find the inverse of the linear transformation Y1 Y2 Y3 = = = 2x₁ 2x₁ x1 +4x₂9x3 +3x₂ -7x3 +2x₂ - 4x3
Find the inverse of the linear transformation Y1 Y2 Y3 = = = 2x₁ 2x₁ x1 +4x₂9x3 +3x₂ -7x3 +2x₂ - 4x3
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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Question
![**Problem Statement:**
Find the inverse of the linear transformation given by the following equations:
\[
\begin{align*}
y_1 &= 2x_1 + 4x_2 - 9x_3 \\
y_2 &= 2x_1 + 3x_2 - 7x_3 \\
y_3 &= x_1 + 2x_2 - 4x_3 \\
\end{align*}
\]
**Explanation:**
This problem involves finding the inverse of a linear transformation represented by a system of linear equations. The transformation maps variables \(x_1, x_2, \text{ and } x_3\) to \(y_1, y_2, \text{ and } y_3\). Solving this requires determining the expressions for \(x_1, x_2, \text{ and } x_3\) in terms of \(y_1, y_2, \text{ and } y_3\).
**Solution Steps:**
1. **Matrix Representation:**
- Write the system in matrix form: \(\mathbf{y} = A\mathbf{x}\), where \(\mathbf{x} = [x_1, x_2, x_3]^T\) and \(\mathbf{y} = [y_1, y_2, y_3]^T\).
- Identify matrix \(A\) based on coefficients of the equations.
2. **Inverse Calculation:**
- Calculate the inverse of matrix \(A\), denoted \(A^{-1}\).
- Use the formula \(\mathbf{x} = A^{-1}\mathbf{y}\) to find \(x_1, x_2, \text{ and } x_3\).
3. **Verification:**
- Multiply matrix \(A\) by its inverse \(A^{-1}\) to ensure the product is the identity matrix.
This transformation and its inverse play a crucial role in various applications, such as computer graphics, physics simulations, and more. Understanding the process of finding the inverse is essential for higher-level mathematics and applied science courses.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b2b0246-74c5-427f-bbae-6a253749f017%2F53663ec4-2286-4e40-a91d-1ef3afd10188%2Fafmd3zn_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the inverse of the linear transformation given by the following equations:
\[
\begin{align*}
y_1 &= 2x_1 + 4x_2 - 9x_3 \\
y_2 &= 2x_1 + 3x_2 - 7x_3 \\
y_3 &= x_1 + 2x_2 - 4x_3 \\
\end{align*}
\]
**Explanation:**
This problem involves finding the inverse of a linear transformation represented by a system of linear equations. The transformation maps variables \(x_1, x_2, \text{ and } x_3\) to \(y_1, y_2, \text{ and } y_3\). Solving this requires determining the expressions for \(x_1, x_2, \text{ and } x_3\) in terms of \(y_1, y_2, \text{ and } y_3\).
**Solution Steps:**
1. **Matrix Representation:**
- Write the system in matrix form: \(\mathbf{y} = A\mathbf{x}\), where \(\mathbf{x} = [x_1, x_2, x_3]^T\) and \(\mathbf{y} = [y_1, y_2, y_3]^T\).
- Identify matrix \(A\) based on coefficients of the equations.
2. **Inverse Calculation:**
- Calculate the inverse of matrix \(A\), denoted \(A^{-1}\).
- Use the formula \(\mathbf{x} = A^{-1}\mathbf{y}\) to find \(x_1, x_2, \text{ and } x_3\).
3. **Verification:**
- Multiply matrix \(A\) by its inverse \(A^{-1}\) to ensure the product is the identity matrix.
This transformation and its inverse play a crucial role in various applications, such as computer graphics, physics simulations, and more. Understanding the process of finding the inverse is essential for higher-level mathematics and applied science courses.
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