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Prove that the following two matrices are not row-equivalent


Solution to Linear Algebra Hoffman & Kunze Chapter 1.3 Exercise 1.3.5

Solution: Call the first matrix A and the second matrix B. The matrix A is row-equivalent to A=[100010001] and the matrix B is row-equivalent to B=[101/2013/2000].

By Theorem 3 page 7 AX=0 and AX=0 have the same solutions. Similarly BX=0 and BX=0 have the same solutions. Now if A and B are row-equivalent then A and B are row equivalent. Thus if A and B are row equivalent then AX=0 and BX=0 must have the same solutions. But BX=0 has infinitely many solutions and AX=0 has only the trivial solution (0,0,0). Thus A and B cannot be row-equivalent.


Linearity

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