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M313Lec06

# M313Lec06 - Math 313 Lecture#6 1.5 Elementary Matrices Part...

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Math 313 Lecture #6 § 1.5: Elementary Matrices, Part I Obtaining Equivalent Systems by Matrix Multiplication. Consider a linear system A~x = ~ b for A an m × n matrix. If M is a nonsingular m × m matrix, then A~x = ~ b is equivalent to MA~x = M ~ b. Why? Because if ~x is a solution of A~x = ~ b , then ~x is a solution of MA~x = M ~ b ; on the other hand if ~x is a solution of MA~x = M ~ b , then applying the inverse of M to both sides gives A~x = M - 1 MA~x = M - 1 M ~ b = ~ b. The idea is to choose M so that MA~x = M ~ b is easier to solve than A~x = ~ b , i.e., ﬁnd M so that MA is in row echelon form. Such an M might be found by applying a sequence of “simple enough” nonsingular matrices E 1 ,...,E k to both sides of A~x = ~ b : E k ··· E 1 A~x = E k ··· E 1 ~ b, that is, M = E k ··· E 1 which is nonsingular because each E i is. Well, we know how to use row operations to get a row echelon form. Can each row operation be achieved by matrix multiplication? Examples. Let A = 1 2 3 4 5 6 7 8 9 , and consider the following three matrices obtained from the identity matrix by a single elementary row operation: 1 0 0 0 1 0 0 0 1 R 2 R 3 1 0 0

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M313Lec06 - Math 313 Lecture#6 1.5 Elementary Matrices Part...

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