MathDB

1 sl8

Part of 2012 Mathcenter Contest + Longlist

Problems(1)

matrices with integers

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

11/13/2022
For matrices A=[aij]m×mA=[a_{ij}]_{m \times m} and B=[bij]m×mB=[b_{ij}]_{m \times m} where A,B∈Zm×mA,B \in \mathbb{Z} ^{m \times m} let A≡B(modn)A \equiv B \pmod{n} only if aij≡bij(modn)a_{ij} \equiv b_{ij} \pmod{n} for every i,j∈{1,2,...,m}i,j \in \{ 1,2,...,m \}, that's A−B=nZA-B=nZ for some Z∈Zm×mZ \in \mathbb{Z}^{m \times m}. (The symbol A∈Zm×mA \in \mathbb{Z} ^{m \times m} means that every element in AA is an integer.) Prove that for A∈Zm×mA \in \mathbb{Z} ^{m \times m} there is B∈Zm×mB \in \mathbb{Z} ^{m \times m} , where AB≡I(modn)AB \equiv I \pmod{n } only if (det⁡(A),n)=1(\det (A),n)=1 and find the value of BB in the form of AA where II represents the dimensional identity matrix m×mm \times m.
(PP-nine)
matrixMatriceslinear algebraalgebra