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Line operations on matrix products

  1. Oct 21, 2014 #1
    Can I perform line operations on a matrix product? For example the matrix product ABC and I know that A and C are invertible, can I row reduce to get ICI? Will the product be conserved?
     
    Last edited: Oct 21, 2014
  2. jcsd
  3. Oct 21, 2014 #2

    HallsofIvy

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    "Line operations"? Do you mean "row operations"? The crucial point of row operations is that every row operation is equivalent to multiplying by a specific "elementary matrix", the matrix you get by applying that row operation to the identity operation. That is if row operations will reduce A to an identity matrix, then there exist a sequence of matrices [itex]P_1[/itex], [itex]P_2[/itex], ..., [itex]P_n[/itex] such that [itex]P_nP_{n-1}...P_2P_1A= I[/itex]. Similarly if C is invertible, there exist a sequence of matrices [itex]Q_1[/itex], [itex]Q_2[/itex], ..., [itex]Q_m[/itex] such that [itex]CQ_1Q_2... Q_{m-1}Q_m= I[/itex]. So applying those row corresponding to the "P" matrices, on the left, and those row operations corresponding to the "Q" matrices , on the right, we would get IBI (which I presume you meant, not "ICI") but, obviously, the product is NOT conserved since, in general, ABC is NOT the same as IBI.
     
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