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2008 District Olympiad
2
Romania District Olympiad 2008 - Grade XI
Romania District Olympiad 2008 - Grade XI
Source:
April 10, 2011
linear algebra
matrix
linear algebra unsolved
Problem Statement
Let
A
,
B
∈
M
n
(
R
)
A,B\in \mathcal{M}_n(\mathbb{R})
A
,
B
∈
M
n
(
R
)
. Prove that
rank
A
+
rank
B
≤
n
\text{rank}\ A+\text{rank}\ B\le n
rank
A
+
rank
B
≤
n
if and only if there exists an invertible matrix
X
∈
M
n
(
R
)
X\in \mathcal{M}_n(\mathbb{R})
X
∈
M
n
(
R
)
such that
A
X
B
=
O
n
AXB=O_n
A
XB
=
O
n
.
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