Given a positive integer n, let a2,n a2,n–1 a3,n–2 a3,n–1 a3,n An = An-1,n аn-1,2 ап-1,n-2 аn-1,л-1 ... An,n-2 An,n-1 An,n An,1 An,2 whose a¡j entries with i+j < n are all equal to 0. Conjecture a formula for det(An) and prove it.
Given a positive integer n, let a2,n a2,n–1 a3,n–2 a3,n–1 a3,n An = An-1,n аn-1,2 ап-1,n-2 аn-1,л-1 ... An,n-2 An,n-1 An,n An,1 An,2 whose a¡j entries with i+j < n are all equal to 0. Conjecture a formula for det(An) and prove it.
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
![Given a positive integer \( n \), let
\[
A_n =
\begin{bmatrix}
0 & 0 & \cdots & 0 & 0 & a_{1,n} \\
0 & 0 & \cdots & 0 & a_{2,n-1} & a_{2,n} \\
0 & 0 & \cdots & a_{3,n-2} & a_{3,n-1} & a_{3,n} \\
\vdots & \vdots & \cdots & \vdots & \vdots & \vdots \\
0 & a_{n-1,2} & \cdots & a_{n-1,n-2} & a_{n-1,n-1} & a_{n-1,n} \\
a_{n,1} & a_{n,2} & \cdots & a_{n,n-2} & a_{n,n-1} & a_{n,n} \\
\end{bmatrix}
\]
whose \( a_{i,j} \) entries with \( i + j \leq n \) are all equal to 0. Conjecture a formula for \( \det(A_n) \) and prove it.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78f85931-ed68-43c6-82f9-075f499bfced%2Fc73ea4d4-b920-4e4a-a27e-71b9edbeed08%2Fx6iqfz4_processed.png&w=3840&q=75)
Transcribed Image Text:Given a positive integer \( n \), let
\[
A_n =
\begin{bmatrix}
0 & 0 & \cdots & 0 & 0 & a_{1,n} \\
0 & 0 & \cdots & 0 & a_{2,n-1} & a_{2,n} \\
0 & 0 & \cdots & a_{3,n-2} & a_{3,n-1} & a_{3,n} \\
\vdots & \vdots & \cdots & \vdots & \vdots & \vdots \\
0 & a_{n-1,2} & \cdots & a_{n-1,n-2} & a_{n-1,n-1} & a_{n-1,n} \\
a_{n,1} & a_{n,2} & \cdots & a_{n,n-2} & a_{n,n-1} & a_{n,n} \\
\end{bmatrix}
\]
whose \( a_{i,j} \) entries with \( i + j \leq n \) are all equal to 0. Conjecture a formula for \( \det(A_n) \) and prove it.
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