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IMC
2009 IMC
3
IMC 2009 Day 2 P3
IMC 2009 Day 2 P3
Source:
July 17, 2014
linear algebra
matrix
algebra
polynomial
IMC
college contests
Problem Statement
Let
A
,
B
∈
M
n
(
C
)
A,B\in \mathcal{M}_n(\mathbb{C})
A
,
B
∈
M
n
(
C
)
be two
n
×
n
n \times n
n
×
n
matrices such that
A
2
B
+
B
A
2
=
2
A
B
A
A^2B+BA^2=2ABA
A
2
B
+
B
A
2
=
2
A
B
A
Prove there exists
k
∈
N
k\in \mathbb{N}
k
∈
N
such that
(
A
B
−
B
A
)
k
=
0
n
(AB-BA)^k=\mathbf{0}_n
(
A
B
−
B
A
)
k
=
0
n
Here
0
n
\mathbf{0}_n
0
n
is the null matrix of order
n
n
n
.
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