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Putnam
1959 Putnam
A6
A6
Part of
1959 Putnam
Problems
(1)
Putnam 1959 A6
Source: Putnam 1959
6/15/2022
Let
m
m
m
and
n
n
n
be integers greater than
1
1
1
and
a
1
,
a
2
,
…
,
a
m
+
1
a_1 ,a_2 ,\ldots, a_{m+1}
a
1
,
a
2
,
…
,
a
m
+
1
be real numbers. Prove that there exist real
n
×
n
n\times n
n
×
n
matrices
A
1
,
A
2
,
…
,
A
m
A_1 ,A_2,\ldots, A_m
A
1
,
A
2
,
…
,
A
m
such that (i)
det
(
A
j
)
=
a
j
\det(A_j) =a_j
det
(
A
j
)
=
a
j
for
j
=
1
,
2
,
…
,
m
j=1,2,\ldots,m
j
=
1
,
2
,
…
,
m
and (ii)
det
(
A
1
+
A
2
+
…
+
A
m
)
=
a
m
+
1
.
\det(A_1 +A_2 +\ldots+A_m)=a_{m+1}.
det
(
A
1
+
A
2
+
…
+
A
m
)
=
a
m
+
1
.
Putnam
matrix
determinant