# Line operations on matrix products

1. Oct 21, 2014

### gummz

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. Oct 21, 2014

### HallsofIvy

Staff Emeritus
"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.