Determinants and equalities

Linear Algebra
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Tolaso J Kos
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Determinants and equalities

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Post by Tolaso J Kos »

Consider the real numbers \( a_{ij} , \; i=1,2, \dots,n-2 \; , \;\; j=1, 2, \dots, n \;\;\; n \geq 3 \) and the determinants

A_k = $$ \begin{vmatrix}
1 &\cdots &1 &1 &\cdots &1 \\
a_{11}&\cdots &a_{1, k-1} &a_{1, k+1} &\cdots & a_{1n}\\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
a_{n-2, 1} &\cdots &a_{n-2, k-1} &a_{n-2, k+1} &\cdots & a_{n-2, n}
\end{vmatrix}$$
Prove that:

a. \( A_1+A_3+A_5+\cdots=A_2+A_4+A_6+\cdots \).

b. For any positive integer \( n \geq 3 \) the following identity holds: \( \displaystyle \sum_{k=1}^{n}\frac{\left ( -1 \right )^k k^2}{\left ( n-k \right )!\left ( n+k \right )!}=0 \) .



D. Andrica
Imagination is much more important than knowledge.
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