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SEEMOUS
2014 SEEMOUS
Problem 3
SEEMOUS 2014(P3)
SEEMOUS 2014(P3)
Source:
April 30, 2015
linear algebra
matrix
Problem Statement
Let
A
∈
M
n
(
C
)
A\in M_n(\mathbb{C})
A
∈
M
n
(
C
)
and
a
∈
C
a\in \mathbb{C}
a
∈
C
such that
A
−
A
∗
=
2
a
I
n
A-A^*=2aI_n
A
−
A
∗
=
2
a
I
n
, where
A
∗
=
(
A
‾
)
T
A^*=(\overline{A})^T
A
∗
=
(
A
)
T
and
I
n
I_n
I
n
is identity matrix. (i) Show that
∣
det
A
∣
≥
∣
a
∣
n
|\det A|\ge |a|^n
∣
det
A
∣
≥
∣
a
∣
n
. (ii) Show that if
∣
det
A
∣
=
∣
a
∣
n
|\det A|=|a|^n
∣
det
A
∣
=
∣
a
∣
n
then
A
=
a
I
n
A=aI_n
A
=
a
I
n
.
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