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SEEMOUS
2013 SEEMOUS
Problem 2
Problem 2
Part of
2013 SEEMOUS
Problems
(1)
over M2(C), M^2+N^2=0 and MN+NM=I, existence of matrix A
Source: SEEMOUS 2013 P2
6/8/2021
Let
M
,
N
∈
M
2
(
C
)
M,N\in M_2(\mathbb C)
M
,
N
∈
M
2
(
C
)
be two nonzero matrices such that
M
2
=
N
2
=
0
2
and
M
N
+
N
M
=
I
2
M^2=N^2=0_2\text{ and }MN+NM=I_2
M
2
=
N
2
=
0
2
and
MN
+
NM
=
I
2
where
0
2
0_2
0
2
is the
2
×
2
2\times2
2
×
2
zero matrix and
I
2
I_2
I
2
the
2
×
2
2\times2
2
×
2
unit matrix. Prove that there is an invertible matrix
A
∈
M
2
(
C
)
A\in M_2(\mathbb C)
A
∈
M
2
(
C
)
such that
M
=
A
(
0
1
0
0
)
A
−
1
and
N
=
A
(
0
0
1
0
)
A
−
1
.
M=A\begin{pmatrix}0&1\\0&0\end{pmatrix}A^{-1}\text{ and }N=A\begin{pmatrix}0&0\\1&0\end{pmatrix}A^{-1}.
M
=
A
(
0
0
1
0
)
A
−
1
and
N
=
A
(
0
1
0
0
)
A
−
1
.
linear algebra
matrix