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FinalTestReviewSpring2011

# FinalTestReviewSpring2011 - MATH 441.001 Instr K Ciesielski...

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MATH 441.001 NAME (print): Instr. K. Ciesielski Spring 2011 FINAL TEST Review Final Test will start with: “Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.)” Remember, That Final Test is comprehensive! This review is based on a combination of the actual tests I have given you, as well as in the in-class reviews. Some problems that I consider especially important to study for the ±nal are marked by read rectangle. 1

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MATH 441.001 NAME (print): Instr. K. Ciesielski Spring 2011 TEST # 1 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Evaluate (a) ° 1 23 ± 1 4 1 = (9 pts) Solution: =[1 8 3] = [ 10] (b) 1 5 1 ° 1 ± = (9 pts) Solution: = 1 5 10 15 12 3 (c) ( AB ) 1 ,i f A 1 = 121 163 012 and B 1 = 111 011 001 . Do not calculate A or B . (13 pts) Solution: ( AB ) 1 = B 1 A 1 = = 096 175 1
Ex. 2. Find the matrix A for which A x y z t = x + y y + z z t x t (10 pts) Solution: A = 110 0 011 0 001 1 100 1 Ex. 3. Using block multiplication, evaluate 1212 3434 5050 (13 pts) Solution: = ° AA ±° BB ± = ° AB + AB AB + AB AB + AB AB + AB ± , where A = ° 12 34 ± and B = ° 50 ± . Since AB = ° ± = ° 74 19 8 ± and AB + AB = ° 19 8 ± + ° 19 8 ± = ° 14 8 38 16 ± , the ±nal answer is 14 8 14 8 38 16 38 16 14 8 14 8 38 16 38 16 Ex. 4. Using Gauss-Jordan elimination, ±nd the inverse of the matrix A = 11 0 01 1 21 0 (13 pts) Solution: [ A : I ]= 11 0100 1010 21 0001 1101 0 0 10 1 0 0 201 10 1 1 101 0 00 1 21 1 10 0 10 1 01 0 20 1 1 21 1 010 2 0 1 001 2 1 1 , so A 1 = 20 1 2 1 1 2

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Ex. 5. (a) Use Guassian elimination to represent the linear system A ° x = ° b in an upper triangular form U ° x = ° c , where A = 1 1 1 22 2 333 , ° x = x y z ,and ° b = 2 0 6 . (12 pts) Solution: [ A : ° b ]= 1 1 12 20 6 2 ° r 1 1 1 040 4 3336 3 ° r 1 1 1 4 0660 3 2 ° r 2 1 1 4 0066 So, in U ° x = ° c we have U = 1 1 1 006 and ° c = 2 4 6 .
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FinalTestReviewSpring2011 - MATH 441.001 Instr K Ciesielski...

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