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VTRMC
1979 VTRMC
3
3
Part of
1979 VTRMC
Problems
(1)
1979 VTRMC #3
Source:
8/8/2018
Let
A
A
A
be an
n
×
n
n\times n
n
×
n
nonsingular matrix with complex elements, and let
A
‾
\overline{A}
A
be its complex conjugate. Let
B
=
A
A
‾
+
I
B = A\overline{A}+I
B
=
A
A
+
I
, where
I
I
I
is the
n
×
n
n\times n
n
×
n
identity matrix. (a) Prove or disprove:
A
−
1
B
A
=
B
‾
A^{-1}BA = \overline{B}
A
−
1
B
A
=
B
. (b) Prove or disprove: the determinant of
A
A
‾
+
I
A\overline{A}+I
A
A
+
I
is real.