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Romania District Olympiad 2004 - Grade XI

Source:

April 10, 2011

linear algebramatrixcalculusderivativealgebrapolynomiallinear algebra unsolved

Problem Statement

Let

A=(a_{ij})\in \mathcal{M}_p(\mathbb{C})

such that

a_{12}=a_{23}=\ldots=a_{p-1,p}=1

and

a_{ij}=0

for any other entry.
a)Prove that

A^{p-1}\neq O_p

and

A^p=O_p

.
b)If

X\in \mathcal{M}_{p}(\mathbb{C})

and

AX=XA

, prove that there exist

a_1,a_2,\ldots,a_p\in \mathbb{C}

such that:

X=\left( \begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_p \\ 0 & a_1 & a_2 & \ldots & a_{p-1} \\ 0 & 0 & a_1 & \ldots & a_{p-2} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & a_1 \end{array} \right)

c)If there exist

B,C\in \mathcal{M}_p(\mathbb{C})

such that

(I_p+A)^n=B^n+C^n,\ (\forall)n\in \mathbb{N}^*

, prove that

B=O_p

or

C=O_p

.

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