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Putnam
1991 Putnam
A2
M^2+N^2 invertible, M^3=N^3, M^2N=N^2M
M^2+N^2 invertible, M^3=N^3, M^2N=N^2M
Source: Putnam 1991 A2
August 20, 2021
matrix
linear algebra
Problem Statement
M
M
M
and
N
N
N
are real unequal
n
×
n
n\times n
n
×
n
matrices satisfying
M
3
=
N
3
M^3=N^3
M
3
=
N
3
and
M
2
N
=
N
2
M
M^2N=N^2M
M
2
N
=
N
2
M
. Can we choose
M
M
M
and
N
N
N
so that
M
2
+
N
2
M^2+N^2
M
2
+
N
2
is invertible?
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