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Romania National Olympiad 2011 - Grade XI - problem 4
Romania National Olympiad 2011 - Grade XI - problem 4
Source:
April 19, 2011
linear algebra
linear algebra unsolved
Problem Statement
Let
A
,
B
∈
M
2
(
C
)
A\, ,\, B\in\mathcal{M}_2(\mathbb{C})
A
,
B
∈
M
2
(
C
)
so that :
A
2
+
B
2
=
2
A
B
A^2+B^2=2AB
A
2
+
B
2
=
2
A
B
. a) Prove that :
A
B
=
B
A
AB=BA
A
B
=
B
A
. b) Prove that :
tr
(
A
)
=
tr
(
B
)
\text{tr}\, (A)=\text{tr}\, (B)
tr
(
A
)
=
tr
(
B
)
.
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