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2023 IMC
2
Matrix equations
Matrix equations
Source: 2023 IMC #2
August 2, 2023
linear algebra
matrix
IMC
IMC 2023
Problem Statement
Let
A
A
A
,
B
B
B
and
C
C
C
be
n
×
n
n \times n
n
×
n
matrices with complex entries satisfying
A
2
=
B
2
=
C
2
and
B
3
=
A
B
C
+
2
I
.
A^2=B^2=C^2 \text{ and } B^3 = ABC + 2I.
A
2
=
B
2
=
C
2
and
B
3
=
A
BC
+
2
I
.
Prove that
A
6
=
I
A^6=I
A
6
=
I
.
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