MathDB

matrices with integers

Source: Mathcenter Contest / Oly - Thai Forum 2012 (R1) p1 sl-8 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

November 13, 2022

matrixMatriceslinear algebraalgebra

Problem Statement

For matrices

A=[a_{ij}]_{m \times m}

and

B=[b_{ij}]_{m \times m}

where

A,B \in \mathbb{Z} ^{m \times m}

let

A \equiv B \pmod{n}

only if

a_{ij} \equiv b_{ij} \pmod{n}

for every

i,j \in \{ 1,2,...,m \}

, that's

A-B=nZ

for some

Z \in \mathbb{Z}^{m \times m}

. (The symbol

A \in \mathbb{Z} ^{m \times m}

means that every element in

A

is an integer.) Prove that for

A \in \mathbb{Z} ^{m \times m}

there is

B \in \mathbb{Z} ^{m \times m}

, where

AB \equiv I \pmod{n }

only if

(\det (A),n)=1

and find the value of

B

in the form of

A

where

I

represents the dimensional identity matrix

m \times m

.
(PP-nine)

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