Finding an invertible matrix by a given relation

Question

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?

Answer 1

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.