Problem 2(*). Find [1 1 1 2 1 1 1 3 4 5 det 1 4 9 16 25 1 8 27 64 125 1 16 81 256 625 using Vandermonde's formula and using the usual definition of determinant.
Problem 2(*). Find [1 1 1 2 1 1 1 3 4 5 det 1 4 9 16 25 1 8 27 64 125 1 16 81 256 625 using Vandermonde's formula and using the usual definition of determinant.
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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May you answer the second question with the asterik?
![**Problem 1.**
The following determinant was introduced by Alexandre-Theophile Vandermonde. Consider distinct real numbers \(a_0, \ldots, a_n\). We define the \((n+1) \times (n+1)\) matrix
\[
A = \begin{bmatrix}
1 & 1 & \cdots & 1 \\
a_0 & a_1 & \cdots & a_n \\
a_0^2 & a_1^2 & \cdots & a_n^2 \\
\vdots & \vdots & \ddots & \vdots \\
a_0^n & a_1^n & \cdots & a_n^n
\end{bmatrix}.
\]
Vandermonde showed that \(\det(A) = \prod_{i > j} (a_i - a_j)\), the product of all differences \(a_i - a_j\), where \(i\) exceeds \(j\).
(a) Verify this formula in the case of \(n = 1\).
(b) Suppose the Vandermonde formula holds for \(n - 1\). You are asked to demonstrate it for \(n\). Consider the function
\[
f(t) = \det \begin{bmatrix}
1 & 1 & \cdots & 1 & 1 \\
a_0 & a_1 & \cdots & a_{n-1} & t \\
a_0^2 & a_1^2 & \cdots & a_{n-1}^2 & t^2 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
a_0^n & a_1^n & \cdots & a_{n-1}^n & t^n
\end{bmatrix}.
\]
Explain why \(f(t)\) is a polynomial of \(n\)-th degree. Find the coefficient \(k\) of \(t^n\) using Vandermonde’s formula for \(a_0, \ldots, a_{n-1}\). Explain why \(f(a_0) = f(a_1) = \cdots = f(a_{n-1}) = 0\). Conclude that \(f(t) = k(t -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5fafa0a8-e885-4f45-9f8c-34c617df8f38%2F70d09f13-1a3a-40d2-9f4f-0d3441749571%2Fx0ykc4_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 1.**
The following determinant was introduced by Alexandre-Theophile Vandermonde. Consider distinct real numbers \(a_0, \ldots, a_n\). We define the \((n+1) \times (n+1)\) matrix
\[
A = \begin{bmatrix}
1 & 1 & \cdots & 1 \\
a_0 & a_1 & \cdots & a_n \\
a_0^2 & a_1^2 & \cdots & a_n^2 \\
\vdots & \vdots & \ddots & \vdots \\
a_0^n & a_1^n & \cdots & a_n^n
\end{bmatrix}.
\]
Vandermonde showed that \(\det(A) = \prod_{i > j} (a_i - a_j)\), the product of all differences \(a_i - a_j\), where \(i\) exceeds \(j\).
(a) Verify this formula in the case of \(n = 1\).
(b) Suppose the Vandermonde formula holds for \(n - 1\). You are asked to demonstrate it for \(n\). Consider the function
\[
f(t) = \det \begin{bmatrix}
1 & 1 & \cdots & 1 & 1 \\
a_0 & a_1 & \cdots & a_{n-1} & t \\
a_0^2 & a_1^2 & \cdots & a_{n-1}^2 & t^2 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
a_0^n & a_1^n & \cdots & a_{n-1}^n & t^n
\end{bmatrix}.
\]
Explain why \(f(t)\) is a polynomial of \(n\)-th degree. Find the coefficient \(k\) of \(t^n\) using Vandermonde’s formula for \(a_0, \ldots, a_{n-1}\). Explain why \(f(a_0) = f(a_1) = \cdots = f(a_{n-1}) = 0\). Conclude that \(f(t) = k(t -
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