MathDB

Modular Matrices

Source: 2018 VTRMC P2

January 8, 2023

VTRMCcollege contestsmatrixmodular arithmeticlinear algebra

Problem Statement

Let

A, B \in M_6 (\mathbb{Z} )

such that

A \equiv I \equiv B \text{ mod }3

and

A^3 B^3 A^3 = B^3

. Prove that

A = I

. Here

M_6 (\mathbb{Z} )

indicates the

6

by

6

matrices with integer entries,

I

is the identity matrix, and

X \equiv Y \text{ mod }3

means all entries of

X-Y

are divisible by

3

.

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