MathDB

determinant of matrix sum

Source: VTRMC 2010 P1

May 17, 2021

linear algebramatrix

Problem Statement

Let

d

be a positive integer and let

A

be a

d\times d

matrix with integer entries. Suppose

I+A+A_2+\ldots+A_{100}=0

(where

I

denotes the identity

d\times d

matrix, and

0

denotes the zero matrix, which has all entries

0

). Determine the positive integers

n\le100

for which

A_n+A_{n+1}+\ldots+A_{100}

has determinant

\pm1

.

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