# Given the following matrix A, find an invertible matrix U so that A is equal to UR, when R is the reduced row-echelon form of A:

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### 1 Answer

Given `A=([0,0,3,-2],[0,0,3,-3],[1,-2,3,-2])` , find an invertible matrix U such that A=UR where R is the reduced row echelon form of A:

Using Gaussian reduction the reduced row echelon form of A is :

`"rref"A=([1,-2,0,0],[0,0,1,0],[0,0,0,1])`

U has to be a 3x3 matrix. Then:

`([a,b,c],[d,e,f],[g,h,i])([1,-2,0,0],[0,0,1,0],[0,0,0,1])=([0,0,3,-2],[0,0,3,-3],[1,-2,3,-2])`

Using matrix multiplication we find a=0,b=3,c=-2,d=0,e=3,f=-3,g=1,h=3,i=-2 so:

`U=([-1,0,1],[1,-2/3,0],[1,-1,0])`

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You can compute U directly by using Gaussian reduction on the augmented matrix A|I:

Reduce `([0,0,3,-2,|,1,0,0],[0,0,3,-3,|,0,1,0],[1,-2,3,-2,|,0,0,1])`

The left side will yield the reduced row echelon form for A, while the right side will be U.

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