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2
IMC 2020 Problem 2
IMC 2020 Problem 2
Source: IMC 2020
July 27, 2020
IMC
linear algebra
IMC 2020
Problem Statement
A
,
B
A, B
A
,
B
are
n
×
n
n \times n
n
×
n
matrices such that
rank
(
A
B
−
B
A
+
I
)
=
1.
\text{rank}(AB-BA+I) = 1.
rank
(
A
B
−
B
A
+
I
)
=
1.
Prove that
tr
(
A
B
A
B
)
−
tr
(
A
2
B
2
)
=
1
2
n
(
n
−
1
)
.
\text{tr}(ABAB)-\text{tr}(A^2 B^2) = \frac{1}{2}n(n-1).
tr
(
A
B
A
B
)
−
tr
(
A
2
B
2
)
=
2
1
n
(
n
−
1
)
.
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