Determinants and equalities
- Tolaso J Kos
- Administrator
- Posts: 867
- Joined: Sat Nov 07, 2015 6:12 pm
- Location: Larisa
- Contact:
Determinants and equalities
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
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.
Create an account or sign in to join the discussion
You need to be a member in order to post a reply
Create an account
Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute
Sign in
Who is online
Users browsing this forum: No registered users and 12 guests