(Solution) - a Suppose A B are m x n matrices such that -(2025 Original AI-Free Solution)

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(a) Suppose A.B are m x n matrices such that ker A = ker B. Prove that there is a nonsingular m × m matrix M such that MA = B.
(b) Use this to conclude that if Ax = b and Bx = c have the same solutions then they are equivalent linear systems, i.e., one can be obtained from the other by a sequence of elementary row operations.