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Let M be a matrix of integer coefficients. Prove that M-1 exists and has integer coefficients if and only if det M = +1. Let {21,..., zn} be any basis of Z". Let M be an n xn matrix of integer coefficients, with

Let M be a matrix of integer coefficients. Prove that M-1 exists and has integer coefficients if and only if det M = +1. Let {21,..., zn} be any basis of Z". Let M be an n xn matrix of integer coefficients, with

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let M be a matrix of integer coefficients. Prove that M-1 exists and has integer coefficients +1. if and only if det M = +1. Let {21, ..., žn} be any basis of Z". Let M be an n x rn matrix of integer coefficients, with det M = +1. Show that {21M,..., Zn M} are again a basis of Zn. Prove that every Abelian group that is finitely generated has countably many elements.
Transcribed Image Text:Let M be a matrix of integer coefficients. Prove that M-1 exists and has integer coefficients +1. if and only if det M = +1. Let {21, ..., žn} be any basis of Z". Let M be an n x rn matrix of integer coefficients, with det M = +1. Show that {21M,..., Zn M} are again a basis of Zn. Prove that every Abelian group that is finitely generated has countably many elements.
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