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SEEMOUS
2024 SEEMOUS
P2
P2
Part of
2024 SEEMOUS
Problems
(1)
Freshman's dream for matrices
Source: SEEMOUS 2024, Problem 2
4/16/2024
Let
A
,
B
∈
M
n
(
R
)
A,B\in\mathcal{M}_n(\mathbb{R})
A
,
B
∈
M
n
(
R
)
two real, symmetric matrices with nonnegative eigenvalues. Prove that
A
3
+
B
3
=
(
A
+
B
)
3
A^3+B^3=(A+B)^3
A
3
+
B
3
=
(
A
+
B
)
3
if and only if
A
B
=
O
n
AB=O_n
A
B
=
O
n
.
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