Let A and B be two different matrices of the same size, both in reduced row-echelon

Chapter 3, Problem 86

(choose chapter or problem)

Let A and B be two different matrices of the same size, both in reduced row-echelon form. Show that the kernelsof A and B are different. (Hint: Focus on the first column in which the two matrices differ, say, the kth columns aic and bk of A and B, respectively. Explain why at least one of the columns a* and bk fails to contain a leading 1. Thus, reversing the roles of matrices A and B if necessary, we can assume that a* does not contain a leading 1. We can write ak as a linear combination of preceding columns and use this representation to construct a vector in the kernel of A. Show that this vector fails to be in the kernel of B. Use Exercises 84 and 85 as a guide.)