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IMC
1999 IMC
1
1
Part of
1999 IMC
Problems
(2)
Easy one
Source: IMC 1999 day 1 problem 1
11/19/2005
a) Show that
∀
n
∈
N
0
,
∃
A
∈
R
n
×
n
:
A
3
=
A
+
I
\forall n \in \mathbb{N}_0, \exists A \in \mathbb{R}^{n\times n}: A^3=A+I
∀
n
∈
N
0
,
∃
A
∈
R
n
×
n
:
A
3
=
A
+
I
. b) Show that
det
(
A
)
>
0
,
∀
A
\det(A)>0, \forall A
det
(
A
)
>
0
,
∀
A
fulfilling the above condition.
linear algebra
matrix
algebra
polynomial
complex numbers
Trivial
Source: IMC 1999 day 2 problem 1
11/19/2005
Let
R
R
R
be a ring where
∀
a
∈
R
:
a
2
=
0
\forall a\in R: a^2=0
∀
a
∈
R
:
a
2
=
0
. Prove that
a
b
c
+
a
b
c
=
0
abc+abc=0
ab
c
+
ab
c
=
0
for all
a
,
b
,
c
∈
R
a,b,c\in R
a
,
b
,
c
∈
R
.
superior algebra
superior algebra solved