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SEEMOUS
2019 SEEMOUS
3
Ranks of matrices
Ranks of matrices
Source: SEEMOUS 2019, problem 3
March 17, 2019
linear algebra
college contests
Problem Statement
Let
A
,
B
A,B
A
,
B
be
n
×
n
n\times n
n
×
n
matrices,
n
≥
2
n\geq 2
n
≥
2
, and
B
2
=
B
B^2=B
B
2
=
B
. Prove that:
rank
(
A
B
−
B
A
)
≤
rank
(
A
B
+
B
A
)
\text{rank}\,(AB-BA)\leq \text{rank}\,(AB+BA)
rank
(
A
B
−
B
A
)
≤
rank
(
A
B
+
B
A
)
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