determinant of the matrix with power series element
by jokerjoestar, Apr 3, 2025, 4:31 PM
Given the function
![\[
f_k(x) = 1 + 2x + 3x^2 + \dots + (k+1)x^k,
\]](//latex.artofproblemsolving.com/f/3/5/f35ea75e6415a63a40fb2c3c95fa5801e6fb891f.png)
show that
![\[
\begin{vmatrix}
f_0(1) & f_1(1) & f_2(1) & \dots & f_{2023}(1) \\
f_0(2) & f_1(2) & f_2(2) & \dots & f_{2023}(2) \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
f_0(2024) & f_1(2024) & f_2(2024) & \dots & f_{2023}(2024)
\end{vmatrix}
= \prod_{k=1}^{2024} k!.
\]](//latex.artofproblemsolving.com/4/7/f/47ff41069b9444fe794076787b8890352929047a.png)
![\[
f_k(x) = 1 + 2x + 3x^2 + \dots + (k+1)x^k,
\]](http://latex.artofproblemsolving.com/f/3/5/f35ea75e6415a63a40fb2c3c95fa5801e6fb891f.png)
show that
![\[
\begin{vmatrix}
f_0(1) & f_1(1) & f_2(1) & \dots & f_{2023}(1) \\
f_0(2) & f_1(2) & f_2(2) & \dots & f_{2023}(2) \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
f_0(2024) & f_1(2024) & f_2(2024) & \dots & f_{2023}(2024)
\end{vmatrix}
= \prod_{k=1}^{2024} k!.
\]](http://latex.artofproblemsolving.com/4/7/f/47ff41069b9444fe794076787b8890352929047a.png)
be all of the extrem points of 
.
An interesting limit
by Alphaamss, Apr 3, 2025, 9:16 AM
Suppose 
I can prove that the sequence 
is convergent by monotone bounded convergence theorem.
Is there any method to compute the limit of, or give the asymptotic representation of ? Any help and hints will welcome!
I can prove that the sequence 
is convergent by monotone bounded convergence theorem.Is there any method to compute the limit of
, or give the asymptotic representation of ? Any help and hints will welcome!
Proving AB-BA is singular from given conditions
by Ciobi_, Apr 2, 2025, 2:04 PM
Let 
be two matrices such that 
. Prove that:
a) if
is odd, then 
;
b) if
, then .
be two matrices such that 
. Prove that:a) if
is odd, then 
;b) if
, then .Easy matrix equation involving invertibility
by Ciobi_, Apr 2, 2025, 1:46 PM
Let be a positive integer, and 
be two complex numbers such that 
and 
, for any 
. The matrices satisfy the relation 
. Prove that 
and 
are invertible.
be a positive integer, and 
be two complex numbers such that 
and 
, for any 
. The matrices satisfy the relation 
. Prove that 
and 
are invertible.
Matrix problem
by hef4875, Mar 26, 2025, 9:49 AM
The matrix 
is defined by the conditions
and 
for a set of indices 
.
Prove that there do not exist nonzero matrices
satisfying the equation
![\[
(I_p + A)^n = B^n + C^n.
\]](//latex.artofproblemsolving.com/2/d/7/2d75d47203353160ccac356251659bd16a4cb5c8.png)
is a postive integer.
is defined by the conditions
and 
for a set of indices 
.Prove that there do not exist nonzero matrices
satisfying the equation![\[
(I_p + A)^n = B^n + C^n.
\]](http://latex.artofproblemsolving.com/2/d/7/2d75d47203353160ccac356251659bd16a4cb5c8.png)
is a postive integer.This post has been edited 2 times. Last edited by hef4875, Mar 26, 2025, 9:51 AM
Putnam 2008 A5
by Kent Merryfield, Dec 8, 2008, 12:23 AM
Let 
be an integer. Let 
and 
be polynomials with real coefficients such that the points 
in 
are the vertices of a regular 
-gon in counterclockwise order. Prove that at least one of and has degree greater than or equal to 
be an integer. Let 
and 
be polynomials with real coefficients such that the points 
in 
are the vertices of a regular 
-gon in counterclockwise order. Prove that at least one of and has degree greater than or equal to 
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