Math 20FFirst Midterm SolutionsJanuary 27,2003The two versions are nearly the same and problems 1 and 3 have been interchanged.The solutions here are for version A with notes on changes for B.1. (#3 in version B) (a)AB=BA=I. I mentioned in class thatAB=IorBA=Iissufficient, so either “AB=I” and “BA=I” are also acceptable.(b) transpose2. There are many ways to convert a matrix to row echelon form. I’ll choose one way.Rn means row n. For version A:10-100001110-12203flflflflflflfl10000001→10-10110-100010005flflflflflflfl10000001add (-2)×(R3) to R4then switch R2 and R3→10-10011-100010000flflflflflflfl10-100001subtract R1 from R2add (-5)×(R3) to R4Thus (a) has solutions; e.g., (1-1 0 0)T; however (b) does not have a solution becausethe last row of the row echelon form augmented matrix is inconsistent.For version B:11-102230100-10010flflflflflflfl00100100→11-1000500-11-10010flfl
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