Line operations on matrix products

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SUMMARY

Line operations, or row operations, can be performed on matrix products, specifically when dealing with invertible matrices A and C in the product ABC. The process involves multiplying by elementary matrices corresponding to the row operations applied to the identity matrix. While it is possible to reduce A and C to identity matrices, the product ABC is not conserved, meaning that the result of the row operations does not yield the same matrix as the original product. The correct notation for the resulting matrix after applying these operations is IBI, not ICI.

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  • Understanding of matrix multiplication and properties
  • Familiarity with elementary matrices and their role in row operations
  • Knowledge of invertible matrices and their significance in linear algebra
  • Basic concepts of row reduction techniques in matrix theory
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Students and professionals in mathematics, particularly those studying linear algebra, matrix theory, and anyone interested in understanding the implications of row operations on matrix products.

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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?
 
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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 P_1, P_2, ..., P_n such that P_nP_{n-1}...P_2P_1A= I. Similarly if C is invertible, there exist a sequence of matrices Q_1, Q_2, ..., Q_m such that CQ_1Q_2... Q_{m-1}Q_m= I. 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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