Given an n xn-matrix A over a field K, the following properties hold:(1) For every column vector b, there is a unique column vector z such that Ax = b iff theonly solution to Ax = 0 is the trivial vector x = 0, iff det(A) # 0.(2) If det (A) #0, the unique solution of Ax=b is given by the expressionsdet (A¹,..., A-¹, b, Aj+¹,Ij:...,A")=det (A¹,..., Ai-¹, Ai, Aj+¹,…, A¹) ¹known as Cramer's rules.(3) The system of linear equations Ax = 0 has a nonzero solution iff det (A) = 0.

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Answered: Given an n xn-matriz A over a field K…

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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Given an n xn-matriz A over a field K ry column vector b, there is a unique column v lution to Ar=0 is the trivial vector x = 0, iff I (A) #0, the unique solution of Ar= b is