Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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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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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