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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

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

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices
Problem 29EQ
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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 the only 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 expressions det (A¹,..., A-¹, b, Aj+¹, Ij: ...,A") = det (A¹,..., Ai-¹, Ai, Aj+¹,…, A¹) ¹ known as Cramer's rules. (3) The system of linear equations Ax = 0 has a nonzero solution iff det (A) = 0.
Transcribed Image Text: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 the only 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 expressions det (A¹,..., A-¹, b, Aj+¹, Ij: ...,A") = det (A¹,..., Ai-¹, Ai, Aj+¹,…, A¹) ¹ known as Cramer's rules. (3) The system of linear equations Ax = 0 has a nonzero solution iff det (A) = 0.
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