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SEEMOUS
2010 SEEMOUS
Problem 3
Problem 3
Part of
2010 SEEMOUS
Problems
(1)
M2(R), existence of matrices satisfying conditions
Source: SEEMOUS 2010 P3
6/17/2021
Denote by
M
2
(
R
)
\mathcal M_2(\mathbb R)
M
2
(
R
)
the set of all
2
×
2
2\times2
2
×
2
matrices with real entries. Prove that:a) for every
A
∈
M
2
(
R
)
A\in\mathcal M_2(\mathbb R)
A
∈
M
2
(
R
)
there exist
B
,
C
∈
M
2
(
R
)
B,C\in\mathcal M_2(\mathbb R)
B
,
C
∈
M
2
(
R
)
such that
A
=
B
2
+
C
2
A=B^2+C^2
A
=
B
2
+
C
2
; b) there do not exist
B
,
C
∈
M
2
(
R
)
B,C\in\mathcal M_2(\mathbb R)
B
,
C
∈
M
2
(
R
)
such that
(
0
1
1
0
)
=
B
2
+
C
2
\begin{pmatrix}0&1\\1&0\end{pmatrix}=B^2+C^2
(
0
1
1
0
)
=
B
2
+
C
2
and
B
C
=
C
B
BC=CB
BC
=
CB
.
matrix
linear algebra