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A fatal linear algebra problem by M. Andronache
A fatal linear algebra problem by M. Andronache
Source: RomNO 2003, grade xi, p.2
August 27, 2019
Matrices
andronache
linear algebra
algebra
Problem Statement
Let be eight real numbers
1
≤
a
1
<
a
2
<
a
3
<
a
4
,
x
1
<
x
2
<
x
3
<
x
4
.
1\le a_1< a_2< a_3< a_4,x_1<x_2<x_3<x_4.
1
≤
a
1
<
a
2
<
a
3
<
a
4
,
x
1
<
x
2
<
x
3
<
x
4
.
Prove that
∣
a
1
x
1
a
1
x
2
a
1
x
3
a
1
x
4
a
2
x
1
a
2
x
2
a
2
x
3
a
2
x
4
a
3
x
1
a
3
x
2
a
3
x
3
a
3
x
4
a
4
x
1
a
4
x
2
a
4
x
3
a
4
x
4
∣
>
0.
\begin{vmatrix}a_1^{x_1} & a_1^{x_2} & a_1^{x_3} & a_1^{x_4} \\ a_2^{x_1} & a_2^{x_2} & a_2^{x_3} & a_2^{x_4} \\ a_3^{x_1} & a_3^{x_2} & a_3^{x_3} & a_3^{x_4} \\ a_4^{x_1} & a_4^{x_2} & a_4^{x_3} & a_4^{x_4} \\ \end{vmatrix} >0.
a
1
x
1
a
2
x
1
a
3
x
1
a
4
x
1
a
1
x
2
a
2
x
2
a
3
x
2
a
4
x
2
a
1
x
3
a
2
x
3
a
3
x
3
a
4
x
3
a
1
x
4
a
2
x
4
a
3
x
4
a
4
x
4
>
0.
Marian Andronache, Ion Savu
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