Answer to Question #24293 in Linear Algebra for Jacob Milne
operation is in one to one correspondence with left matrix multiplying by
elementary, and they are invertible, so matrix A and B are equivalent by
viewing that B=E1E2…EnA, and Ei areelementary matrices. Thus systems Ax=0 and Bx=0 have the same solutions and by Rouché–Capelli theorem rank(B) = rank(A) =rank(A|0). If B has N non-zero rows, then N=rank(A) and A is invertible, so x=0. Then B has to have R fewer than Nnon-zero rows, then we canchoose N-R nonzero values of x components, and obtain the whole nonzero x.
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