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3 - Equivalence and Inverses

3 - Equivalence and Inverses - Math 1b Row-equivalence...

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Math 1b — Row-equivalence; matrix inverses January 7, 2011 Recall that matrices A and B are row-equivalent when one can be obtained from the other by a sequence of elementary row operations. An elementary row operation on a matrix M gives us a matrix whose rows M whose rows are linear combinations of the rows of M . Since elementary row operations can be ‘undone’ by other elementary row operations, the rows of M are linear combinations of the rows of M . It follows that if A and B are row-equivalent, then the rows of A are linear combinations of the rows of B , and the rows of B are linear combinations of the rows of A . (Later, we will say that A and B have the same row space .) To review one item from the handout on matrix multiplication, recall that given matrices A and B , the equation A = CB holds for some matrix C if and only if the rows of A are linear combinations of the rows of B . For example, a 11 a 12 a 13 a 21 a 22 a 23 = 2 3 4 5 6 7 8 9 b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 b 41 b 42 b
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