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South Korea USCM
2018 Korea USCM
4
4
Part of
2018 Korea USCM
Problems
(1)
Find determinant of matrix obtained by two permuations
Source: 2018 South Korea USCM P4
8/14/2020
n
≥
2
n\geq 2
n
≥
2
is a given integer. For two permuations
(
α
1
,
⋯
,
α
n
)
(\alpha_1,\cdots,\alpha_n)
(
α
1
,
⋯
,
α
n
)
and
(
β
1
,
⋯
,
β
n
)
(\beta_1,\cdots,\beta_n)
(
β
1
,
⋯
,
β
n
)
of
1
,
⋯
,
n
1,\cdots,n
1
,
⋯
,
n
, consider
n
×
n
n\times n
n
×
n
matrix
A
=
(
a
i
j
)
1
≤
i
,
j
≤
n
A= \left(a_{ij} \right)_{1\leq i,j\leq n}
A
=
(
a
ij
)
1
≤
i
,
j
≤
n
defined by
a
i
j
=
(
1
+
α
i
β
j
)
n
−
1
a_{ij} = (1+\alpha_i \beta_j )^{n-1}
a
ij
=
(
1
+
α
i
β
j
)
n
−
1
. Find every possible value of
det
(
A
)
\det(A)
det
(
A
)
.
linear algebra
matrix
determinant
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