# Finding an invertible matrix by a given relation

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?

• 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. – KeiOh May 8 '20 at 13:59
• 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 – Darkenin May 8 '20 at 14:06
• Well you did not give an explicit description of $A$, so it is hard to help you further. – KeiOh May 9 '20 at 0:44

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.