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Practice Midterm 1-1 ans

# Practice Midterm 1-1 ans - Math 20F First Midterm Solutions...

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Math 20F First Midterm Solutions January 27,2003 The 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 that AB = I or BA = I is sufficient, so either “ AB = I ” and “ BA = I ” are also acceptable. (b) transpose 2. There are many ways to convert a matrix to row echelon form. I’ll choose one way. Rn means row n. For version A: 1 0 - 1 0 0 0 0 1 1 1 0 - 1 2 2 0 3 fl fl fl fl fl fl fl 1 0 0 0 0 0 0 1 1 0 - 1 0 1 1 0 - 1 0 0 0 1 0 0 0 5 fl fl fl fl fl fl fl 1 0 0 0 0 0 0 1 add ( - 2) × (R3) to R4 then switch R2 and R3 1 0 - 1 0 0 1 1 - 1 0 0 0 1 0 0 0 0 fl fl fl fl fl fl fl 1 0 - 1 0 0 0 0 1 subtract R1 from R2 add ( - 5) × (R3) to R4 Thus (a) has solutions; e.g., (1 - 1 0 0) T ; however (b) does not have a solution because the last row of the row echelon form augmented matrix is inconsistent. For version B: 1 1 - 1 0 2 2 3 0 1 0 0 - 1 0 0 1 0 fl fl fl fl fl fl fl 0 0 1 0 0 1 0 0 1 1 - 1 0 0 0 5 0 0 - 1 1 - 1 0 0 1 0 fl fl
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