MathDB
Problems
Contests
National and Regional Contests
Romania Contests
District Olympiad
2011 District Olympiad
2
Romania District Olympiad 2011 - Grade XI
Romania District Olympiad 2011 - Grade XI
Source:
March 12, 2011
linear algebra
matrix
algebra
polynomial
linear algebra unsolved
Problem Statement
Consider the matrices
A
∈
M
m
,
n
(
C
)
A\in \mathcal{M}_{m,n}(\mathbb{C})
A
∈
M
m
,
n
(
C
)
and
B
∈
M
n
,
m
(
C
)
B\in \mathcal{M}_{n,m}(\mathbb{C})
B
∈
M
n
,
m
(
C
)
with
n
≤
m
n\le m
n
≤
m
. It is given that
rank
(
A
B
)
=
n
\text{rank}(AB)=n
rank
(
A
B
)
=
n
and
(
A
B
)
2
=
A
B
(AB)^2=AB
(
A
B
)
2
=
A
B
. a)Prove that
(
B
A
)
3
=
(
B
A
)
2
(BA)^3=(BA)^2
(
B
A
)
3
=
(
B
A
)
2
. b)Find
B
A
BA
B
A
.
Back to Problems
View on AoPS