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If I have some matrix $A$, which satisfies $A=PU$ where U is the reduced row echelon form matrix and P is some invertible matrix I have to find. How do I go about it? Is there an easier way other than reducing A into row echelon form while doing the elementary operations on an identity matrix and then finding its inverse?

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  • $\begingroup$ If you have no additional information on $A$, I do not see how you could hope for something easier. The row echelon form is an algorithm, and while the $P$ matrix encodes all the transformation you need, you still need to compute them. $\endgroup$ – KeiOh May 8 '20 at 13:59
  • $\begingroup$ I do have the matrix A, and it is said that there is no need to calculate the reduced echelon form in order to get the answer $\endgroup$ – Darkenin May 8 '20 at 14:06
  • $\begingroup$ Well you did not give an explicit description of $A$, so it is hard to help you further. $\endgroup$ – KeiOh May 9 '20 at 0:44
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Here is a way that at least obviates the susequent calculation of $P^{-1}.$ bu instead calculates $P^{-1}$ as you go along. Suppose you calculate the row-reduced form by multiplying $A$ by $P_1,P_2,...,P_{t-1},P_t$ so $$U=P_tP_{t-1}...P_2P_1A$$ so $$A=(P_tP_{t-1}...P_2P_1)^{-1}U$$ and $$P=(P_tP_{t-1}...P_2P_1)^{-1}$$ $$=P_1^{-1}P_2^{-1}...P_{t-1}^{-1}P_t^{-1}$$ $$=IP_1^{-1}P_2^{-1}...P_{t-1}^{-1}P_t^{-1}$$ Thus, while you are calculating $U$ you can simulaneously calculate $P$ by applying column operations, starting with the identity matrix $I.$ For example, if multiplying a matrix on the left by $P_i$ is multiplying the fifth row by 4, then multiplying a matrix on the right by $P_i^{-1}$ is multiplying the fifth column by $\frac{1}{4}.$The other two types of elementary row operations also correspond to columns operations, you can work out the correspondence, which is not exactly what you might think.

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