Fall 2011 Prelim 1 Solutions

# Fall 2011 Prelim 1 Solutions - Math 2940 Prelim 1 Solutions...

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Math 2940, Prelim 1 Solutions Fall 2011 1) A matrix A was reduced to A 0 = 1 0 2 0 1 - 1 0 0 0 by the following row operations in the given order: 2 times Row 1 subtracted from Row 2, 4 times Row 1 added to Row 3, Row 2 multiplied by - 1 2 and ﬁnally Row 2 subtracted from Row 3. (a) (10 points) What was the original matrix A ? (b) (8 points) What matrix multiplication combines all the row operations, in other words for what matrix L is A 0 = LA ? Solution: (a) Working backwards, we add Row 2 to Row 3 to get 1 0 2 0 1 - 1 0 1 - 1 then multiply Row 2 by - 2 to get 1 0 2 0 - 2 2 0 1 - 1 subtract 4 times Row 1 from Row 3 to get 1 0 2 0 - 2 2 - 4 1 - 9 and add 2 times Row 1 to Row 4 to get A = 1 0 2 2 - 2 6 - 4 1 - 9 . (b) We have 1 0 0 0 1 0 0 - 1 1 1 0 0 0 - 1 / 2 0 0 0 1 1 0 0 0 1 0 4 0 1 1 0 0 - 2 1 0 0 0 1 A = 1 0 2 0 1 - 1 0 0 0 so L = 1 0 0 0 1 0 0 - 1 1 1 0 0 0 - 1 / 2 0 0 0 1 1 0 0 0 1 0 4 0 1 1 0 0 - 2 1 0 0 0 1 = 1 0 0 0 - 1 / 2 0 0 1 / 2 1 1 0 0 - 2 1 0 4 0 1 = 1 0 0 1 - 1 / 2 0 3 1 / 2 1 . 2) (16 points) Find all ~x that simultaneously satisfy both of the following two systems. 1 2 1

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Fall 2011 Prelim 1 Solutions - Math 2940 Prelim 1 Solutions...

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