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District Olympiad
2009 District Olympiad
2
Romania District Olympiad 2009 - Grade XI
Romania District Olympiad 2009 - Grade XI
Source:
April 10, 2011
linear algebra
matrix
linear algebra unsolved
Problem Statement
Let
n
∈
N
∗
n\in \mathbb{N}^*
n
∈
N
∗
and a matrix
A
∈
M
n
(
C
)
,
A
=
(
a
i
j
)
1
≤
i
,
j
≤
n
A\in \mathcal{M}_n(\mathbb{C}),\ A=(a_{ij})_{1\le i, j\le n}
A
∈
M
n
(
C
)
,
A
=
(
a
ij
)
1
≤
i
,
j
≤
n
such that:
a
i
j
+
a
j
k
+
a
k
i
=
0
,
(
∀
)
i
,
j
,
k
∈
{
1
,
2
,
…
,
n
}
a_{ij}+a_{jk}+a_{ki}=0,\ (\forall)i,j,k\in \{1,2,\ldots,n\}
a
ij
+
a
jk
+
a
ki
=
0
,
(
∀
)
i
,
j
,
k
∈
{
1
,
2
,
…
,
n
}
Prove that
rank
A
≤
2
\text{rank}\ A\le 2
rank
A
≤
2
.
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