x x²] 4 -1 1 y y². The matrix A is known as a Vandermonde matrix. 1 z z² Vandermonde matrices have many applications in different areas of mathemat- 8.24. Let A = ics. Show that det(A) = (x - y) (y - z) (z − x). at the Vandermonde matrix is invert-
x x²] 4 -1 1 y y². The matrix A is known as a Vandermonde matrix. 1 z z² Vandermonde matrices have many applications in different areas of mathemat- 8.24. Let A = ics. Show that det(A) = (x - y) (y - z) (z − x). at the Vandermonde matrix is invert-
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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![### Vandermonde Matrices and Their Applications
**Problem 8.24.**
Consider the matrix \( A \) defined as follows:
\[
A =
\begin{bmatrix}
1 & x & x^2 \\
1 & y & y^2 \\
1 & z & z^2 \\
\end{bmatrix}
\]
The matrix \( A \) is known as a **Vandermonde matrix**.
Vandermonde matrices have numerous applications in various fields of mathematics, including polynomial interpolation, coding theory, and numerical analysis.
**Tasks:**
1. **Determinant of the Vandermonde Matrix:**
Show that the determinant of \( A \), denoted as \( \text{det}(A) \), is given by:
\[
\text{det}(A) = (x - y)(y - z)(z - x)
\]
2. **Invertibility Conditions:**
Determine the conditions that \( x, y, \text{and} z \) must satisfy for the Vandermonde matrix to be invertible.
**Explanation of Determinant and Invertibility:**
- The determinant of the Vandermonde matrix is a product of the differences between each pair of \( x, y, \text{and} z \).
- For the matrix \( A \) to be invertible, its determinant must be non-zero. Hence, \( x, y, \text{and} z \) must all be distinct; that is:
\[
x \neq y \\
y \neq z \\
z \neq x
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1cc0b2c-4a48-4343-8af7-918206120553%2F97e63c7b-86a9-44d4-8dac-8c12dec88118%2F7psga98_processed.png&w=3840&q=75)
Transcribed Image Text:### Vandermonde Matrices and Their Applications
**Problem 8.24.**
Consider the matrix \( A \) defined as follows:
\[
A =
\begin{bmatrix}
1 & x & x^2 \\
1 & y & y^2 \\
1 & z & z^2 \\
\end{bmatrix}
\]
The matrix \( A \) is known as a **Vandermonde matrix**.
Vandermonde matrices have numerous applications in various fields of mathematics, including polynomial interpolation, coding theory, and numerical analysis.
**Tasks:**
1. **Determinant of the Vandermonde Matrix:**
Show that the determinant of \( A \), denoted as \( \text{det}(A) \), is given by:
\[
\text{det}(A) = (x - y)(y - z)(z - x)
\]
2. **Invertibility Conditions:**
Determine the conditions that \( x, y, \text{and} z \) must satisfy for the Vandermonde matrix to be invertible.
**Explanation of Determinant and Invertibility:**
- The determinant of the Vandermonde matrix is a product of the differences between each pair of \( x, y, \text{and} z \).
- For the matrix \( A \) to be invertible, its determinant must be non-zero. Hence, \( x, y, \text{and} z \) must all be distinct; that is:
\[
x \neq y \\
y \neq z \\
z \neq x
\]
Transcribed Image Text:### Invertibility Condition
**8.24** To be invertible, \( x \), \( y \), and \( z \) must be distinct \((x \ne y \ne z)\).
**Explanation:**
For a function or matrix to be invertible, it is essential that the elements \( x \), \( y \), and \( z \) are unique or distinct from each other. This means none of the values \( x \), \( y \), and \( z \) should be equal. In mathematical notation, this condition is expressed as \( x \ne y \ne z \).
This requirement ensures that the function or matrix can be reversed or inverted, thereby allowing one to retrieve original inputs from the outputs without ambiguity.
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