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Romania National Olympiad
2021 Romania National Olympiad
2
Romania National Olympiad Grade 11 P2
Romania National Olympiad Grade 11 P2
Source:
April 28, 2021
linear algebra
Problem Statement
Let
n
≥
2
n \ge 2
n
≥
2
and
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots , a_n
a
1
,
a
2
,
…
,
a
n
, nonzero real numbers not necessarily distinct. We define matrix
A
=
(
a
i
j
)
1
≤
i
,
j
≤
n
∈
M
n
(
R
)
A = (a_{ij})_{1 \le i,j \le n} \in M_n( \mathbb{R} )
A
=
(
a
ij
)
1
≤
i
,
j
≤
n
∈
M
n
(
R
)
,
a
i
,
j
=
m
a
x
{
a
i
,
a
j
}
a_{i,j} = max \{ a_i, a_j \}
a
i
,
j
=
ma
x
{
a
i
,
a
j
}
,
∀
i
,
j
∈
{
1
,
2
,
…
,
n
}
\forall i,j \in \{ 1,2 , \ldots , n \}
∀
i
,
j
∈
{
1
,
2
,
…
,
n
}
. Show that
r
a
n
k
(
A
)
\mathbf{rank}(A)
rank
(
A
)
=
c
a
r
d
\mathbf{card}
card
{
a
k
∣
k
=
1
,
2
,
…
n
}
\{ a_k | k = 1,2, \ldots n \}
{
a
k
∣
k
=
1
,
2
,
…
n
}
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