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2010 VTRMC
Problem 1
Problem 1
Part of
2010 VTRMC
Problems
(1)
determinant of matrix sum
Source: VTRMC 2010 P1
5/17/2021
Let
d
d
d
be a positive integer and let
A
A
A
be a
d
×
d
d\times d
d
×
d
matrix with integer entries. Suppose
I
+
A
+
A
2
+
…
+
A
100
=
0
I+A+A_2+\ldots+A_{100}=0
I
+
A
+
A
2
+
…
+
A
100
=
0
(where
I
I
I
denotes the identity
d
×
d
d\times d
d
×
d
matrix, and
0
0
0
denotes the zero matrix, which has all entries
0
0
0
). Determine the positive integers
n
≤
100
n\le100
n
≤
100
for which
A
n
+
A
n
+
1
+
…
+
A
100
A_n+A_{n+1}+\ldots+A_{100}
A
n
+
A
n
+
1
+
…
+
A
100
has determinant
±
1
\pm1
±
1
.
linear algebra
matrix