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Putnam
1940 Putnam
B6
Putnam 1940 B6
Putnam 1940 B6
Source: Putnam 1940
February 22, 2022
Putnam
linear algebra
matrix
Matrix determinant
Problem Statement
Prove that the determinant of the matrix
(
a
1
2
+
k
a
1
a
2
a
1
a
3
…
a
1
a
n
a
2
a
1
a
2
2
+
k
a
2
a
3
…
a
2
a
n
…
…
…
…
…
a
n
a
1
a
n
a
2
a
n
a
3
…
a
n
2
+
k
)
\begin{pmatrix} a_{1}^{2}+k & a_1 a_2 & a_1 a_3 &\ldots & a_1 a_n\\ a_2 a_1 & a_{2}^{2}+k & a_2 a_3 &\ldots & a_2 a_n\\ \ldots & \ldots & \ldots & \ldots & \ldots \\ a_n a_1& a_n a_2 & a_n a_3 & \ldots & a_{n}^{2}+k \end{pmatrix}
a
1
2
+
k
a
2
a
1
…
a
n
a
1
a
1
a
2
a
2
2
+
k
…
a
n
a
2
a
1
a
3
a
2
a
3
…
a
n
a
3
…
…
…
…
a
1
a
n
a
2
a
n
…
a
n
2
+
k
is divisible by
k
n
−
1
k^{n-1}
k
n
−
1
and find its other factor.
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