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District Olympiad
2005 District Olympiad
3
Romania District Olympiad 2005 - Grade XI
Romania District Olympiad 2005 - Grade XI
Source:
April 10, 2011
algebra
polynomial
linear algebra
linear algebra unsolved
Problem Statement
a)Let
A
,
B
∈
M
3
(
R
)
A,B\in \mathcal{M}_3(\mathbb{R})
A
,
B
∈
M
3
(
R
)
such that
rank
A
>
rank
B
\text{rank}\ A>\text{rank}\ B
rank
A
>
rank
B
. Prove that
rank
A
2
≥
rank
B
2
\text{rank}\ A^2\ge \text{rank}\ B^2
rank
A
2
≥
rank
B
2
.b)Find the non-constant polynomials
f
∈
R
[
X
]
f\in \mathbb{R}[X]
f
∈
R
[
X
]
such that
(
∀
)
A
,
B
∈
M
4
(
R
)
(\forall)A,B\in \mathcal{M}_4(\mathbb{R})
(
∀
)
A
,
B
∈
M
4
(
R
)
with
rank
A
>
rank
B
\text{rank}\ A>\text{rank}\ B
rank
A
>
rank
B
, we have
rank
f
(
A
)
>
rank
f
(
B
)
\text{rank}\ f(A)>\text{rank}\ f(B)
rank
f
(
A
)
>
rank
f
(
B
)
.
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