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SEEMOUS
2024 SEEMOUS
P4
P4
Part of
2024 SEEMOUS
Problems
(1)
Necessary condition for symmetric matrix
Source: SEEMOUS 2024, Problem 4
4/16/2024
Let
n
∈
N
n\in\mathbb{N}
n
∈
N
,
n
≥
2
n\geq 2
n
≥
2
. Find all values of
k
∈
N
k\in\mathbb{N}
k
∈
N
,
k
≥
1
k\geq 1
k
≥
1
, for which the following statement holds:
"If
A
∈
M
n
(
C
)
is such that
A
k
A
∗
=
A
, then
A
=
A
∗
."
\text{"If }A\in\mathcal{M}_n(\mathbb{C})\text{ is such that }A^kA^*=A\text{, then }A=A^*\text{."}
"If
A
∈
M
n
(
C
)
is such that
A
k
A
∗
=
A
, then
A
=
A
∗
."
(here,
A
∗
A^*
A
∗
denotes the conjugate transpose of
A
A
A
).
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