Test4 171 review

# Test4 171 review - 1. Compute: MATH 17100 1 1 28446 Q11 31...

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Unformatted text preview: 1. Compute: MATH 17100 1 1 28446 Q11 31 3 1 uiz 13 solution 0 2 (a) 2211 40 1. 2 ompute2 1 C1 : MATH 17100 28446 Quiz 13 solution MATH 17100 28446 Quiz 13 solution 31 11 31 11 20 1. C1. C(a) : : ompute ompute 1 2 21 2211 40 31 (a) 3 a) 1 31 ( 2 12 (b) 212 11/04/ 2009 11/04/ 20092009 11/04/ 11 11 1 3 1 (3)(1) 12 12 1 2 1 ( 2)(1) 13 1 1 1 1 2 1 (1)(2) 1 (3)(1) 22 12 1 1 4 (1)(2) ( 2)(1) 00 2 (1)(1) 04 0 (1)(1) 5 0 4 1 212 3111 (b) (b) 3 1 1 y 22 w x 1 12 21 and 2 2. (a) If zwx (b) 3 111 1 (3)(1) (1)(2) (3)(1) (1)(1) 5 4 ( 2)(1) (1)(2) ( 2)(1) (1)(1) 01 (3)(1) (1)(1) 1 (3)(1) (1)(2) (3)(1) (1)(2) (3)(1) (1)(1) 5 4 4 5 u ( 2)(1) (1)(2) ( 2)(1) (1)(1) 4y 0 0 equation. 1 1 ( 2)(1), write each system (1)(1) (1)(2) ( 2)(1) as a matrix 1 v 3y 4z u 4y y 2w x and , write each system as a matrix equation. v4 3 0 4y y z zw w x u y 21 yy . (z If 1 12 wx 2x and v 4 y 4 y4 , write each system as a matrix equation. w and u u 3 x z , write each system as a matrix equation. 2 a) . (a) If nd 4z 2 z w x x av 3vy 3 y 4 z zw y 21w u 40y and z v 11x 3 4 zw u y y 2 12 w w y (b) Write a matrix1equation u u 4 0 and y relating 4 0 and and v x z z 1 11 x x v v 3 3 z4 z 4 1 u 40y 4 0 2u w 1 w From (a),b) Write a matrix equation relating ( and v 3 4z 3 4 1v x 1 x M A T H 17100 28446 Q u i z 14 solu t ion u 11/09/ 2009 ww M A T(H 1Write a28446 equation relatingi z 14 souu t ion Qu l 11/09/ 2009 b) 7100 matrix and and (b) Write a matrix equation relating 0 2 x 1 w 8u 4 4 w 0 y 4 v v 1. FindFindcoefficient matrix and the augmented matrix ofxthe following system of of F 1. therom (a), 2v matrix 4andzthe augmented matrix of the following system the coefficient 1 3 x 3 411x equations: y equations:u u 4 0 0 y 4 0 2 12 w w 4 40 1 From (a), F 2y z 4 z4 w 3 411x 8 2 x rom (a),v 1 v 3 4z 3 411x 2x 2 y z 1 3 2 1x (a) x y 2 z 3 (a) x y 2 z 3 4 w 8 84w x 3y 4z 6 x 3y 4z 2 6 R.Tam 1x 2 1x 22 1 22 11 22 1 22 11 11 2 , Augmented matrix, A | b Coefficient matrix, A 11 2 3 11 2 , Augmented matrix, A | b 11 2 3 R.Tam Coefficient matrix, A 13 4 13 46 13 4 13 46 2. (a) If R.Tam R.Tam (b) z1 2x z 1 y 2z 3 (b) y 2 z 3 3y 4z 6 3y 4z 6 2x 2 0 (c) Coefficient matrix, A (c) Coefficient matrix, A 0 0 1 3 2 0 0 1 0 2 1 4 3 , Augmented matrix, A | b 0 2 , Augmented matrix, A | b 4 1 2 0 1 0 1 2 01 3 4 03 0 2 1 1 3 2 6 4 1 3 6 3y 4z 6 3y 4z 6 2 0 1 20 11 20 1 20 11 1 2 , Augmented matrix, A | b 01 2 3 01 2 , Augmented matrix, A | b 01 2 3 3 4 03 46 03 4 03 46 0 (c) Coefficient matrix, A (c) Coefficient matrix, A 0 2. Write the system of equations represented by the following augmented matrices 2. Write the system of equations represented by the following augmented matrices 1 (a) 2 (a) 1 12 1 12 0 1 20 1 1 0 11 0 2 y2 2x 1 2x 1 xy0 xy0 x Denoting the unknowns x and y, we have Denoting the unknowns x and y, we have 1 1 y x 11 (b) (b) 2 2 22 M A T H 17100 28446 Q u i z 15 sol u t ion 11/11/ 2009 MA T H 17 10 22846 6 84 4 Q i i z s 5 so l n i o n 1 11/9 MTA T H is71000 one4unknown,uQzu15 1ol u t,iou tthe system of equations 11/11/ 20011/ 2009 denoted of 1.here theisow-reduced echelon denotedin, in the system of equations intermediate steps: Find only one unknown, form x the following matrices, showing r only There x 1 x 1 the ow-reduced echelon form of following matrices, showing intermediate steps: Find 1..Find the rrow-reduced echelon form of the the following matrices, showing intermediate steps: (a) 1 1 x2 1 2 11 2 x 2 22 11 2 2 (a) 21x 212 1 12 1 1 11 1 1 2 2 1 1 1 1 2 2 1 1 2 2 1 2 1 1 2 2 1 2 1 12 12 21 21 1 2 1 2 1 1 21 1 2 1 21 1 0 0 1 R.Tam ( R.Tama) 21 21 1 1 11 1 1 0 0 10 10 00 00 3 5 2 2 25 3 3 23 1 0 0 1 0 0 10 10 21 21 00 00 15 13 5 3 1 2 53 33 1 1 (b) 1 1 100 10 11 0 10 11 01 001 0 0 0 1 1 01 01 0 2 0 3 3 1 0 10 0 0 1 1 0 0 110 0 01 00 000 000 1 10 01 0 0 0 5 3 0 00 00 102 1 2 0 0 0 0 0 22 1 (b) 21 21 12 1 1 1 22 2 2 1 1 1 1 11 2 2 1 12 1 11 1 22 1 1 12 1 1 22 21 1 1 1 21 21 1 0 0 12 12 21 21 1 1 1 21 0 3 0 3 21 23 33 3 0 0 1 0 0 2 1 1 1 2 1 1 1 2 12 1 0 0 0 0 10 10 00 00 0 0 23 3 3 10 10 21 21 00 00 11 11 0 0 1 1 010 1 11 0 0 0 0 11 10 10 0 0 000 00 0 1 1 0 0 0 0 (b) 11 11 2 1 2 1 1 1 1 1 11 11 33 3 2. Determine whether or not the following two systems are equivalent: 2. Determine whether or not the following two systems are equivalent: x 2x 1 x x x 0 2. Determine whether or not the following two systems are equivalent: 2x x x 1 x x x 1 1 3 4 1 3 x11 x23 x4 4 0 x 3 x xx x1 x x 0 1 x1 1 4 3 2 4 4 x1 x 2 x 4 1 (I) and 1 x12 2 4x 3 1 x x 31 x x 42 x 14 1 (I) (I) and and x1 x 2 x x 4 xx 0 1x x31 1 2 3 2 4 x1 x 2 x 3 2 1 1 0 2x 11 x xx32 xx 4 4 Ax x1 xx 2 1matrix 1 ugmented x 4 corresponding to (I): 3 4 x1 x 2 x 4 1 0 1 0 0 1 1 1 11 0 1 0 1 1 101 0 1 1 1 1 01 x1 x1 x 3 x 2 x 4 x 3 0 2 x1 x2 1 0 0 1 1 0 1 10 10 1 Augmented matrix corresponding to1(I):0 11 0 1 1 01 1 Augmented matrix corresponding to0(I): 1 0 0 x3 1 1 2 1 0 0 1 1 0 0 1 1 1 11 00 01 0 10 01 1 1 1 21 1 0 1 0 11 0 1 1 1 1 1 0 102 01 2 11 2 0 0 0 1 1 1 1 1 1 1 1 1 0 0 3 3 0 0 1 1 0 0 0 0 2. Determine whether or not the following two systems are equivalent: x1 x1 x3 x1 x3 x2 x4 x2 1 x4 x4 x4 0 1 x1 2 x3 1 (I) and 2 x1 x2 x3 x2 x4 x4 x3 0 1 x1 x1 1 2 1 0 1 10 10 1 10 10 1 10 11 0 1 1 01 10 1 01 10 1 001 11 001 11 001 11 17100 1 28446 0 1 Quiz 16 solution 1 MATH 1 1 1 01 2 1 0 0 2 11/2 6/ 2009 2 1. Find the r1 0of the coefficient matrix as0well2as that of the augmented matrix associated ank 0 2 1 1000 with the given systems of1 equations.0Then solve t1 system of equations1and express the he 010 1 2 010 12 01001 solution as a vector: 001 1 1 001 11 00100 MATH 17100 28446 Quiz 16 solution 11/16/ 2009 3x 0 2 z 0 0 4 y0 4 000 1 1 0001 1 12Findythe z ank of the coefficient matrix as well as that of the augmented matrix associated .x r1 (a) x solutionzas a vector: 2y 4 with the given systems of equations. Then solve the system of equations and express the Augmented matrix corresponding to (I): Augmented matrix,x y 2 z 0 3 3 1(a) 2 x y z 1 1 20 A|b 2 1 x 1 2y z 1 4 0 2 3 1 3 4 9 10 01 1 1 2 3 10 01 10 10 4 0 5 5 12 000 3 0001 A1 2 matrix, ugmented 1 31 20 121 4 10 1 2 10 10 rows rank ( A) A|2, rank ( A1| b)1 3 1from the number of non-zero 1 1 in3the echelon form. b 2 033 9 0 0110 12 14 0 5 5 12 000 3 0001 rank ( A) rank ( A | b) System is inconsistent and no solution exists rank ( A) 2, rank ( A | b) 3 from the number of non-zero rows in the echelon form. 3y 9z 6 rankyA)3 z rank ( A | b) System is inconsistent and no solution exists ( (b) 2 2 y 6 z 3 y4 9 z 6 (b) y 3 z 2 396 132 132 2 y 6z 4 132 132 000 Augmented matrix, A|b 396 132 132 264 132 000 Augmented matrix, A|b 1 3 2 132 000 2 4 132 000 rank ( A) 1, rank ( A | b) 1 from the number6of non-zero rows in the echelon form. Free variable is z, basic variable is y. Solution is z , arbitrary, y , rank ( A) 1, rank ( A | b) 1 from the number of non-zero rows in the echelon form. Free variable is z, basic variable is y. Solution is 23 z arbitrary, y 23 y 23 2 3 MATH 17100 28446 MATH 17100 28446 Quizuiz solution Q 17 17 solution 11/18/ 18/ 2009 1 2009 1. Evaluate thethe following determinant using the the Laplace Expansion Method 1. Evaluate following determinant by by using Laplace Expansion Method MATH 17100 28446 Quiz 17 olution MATH (17100 row expansion). Which rowscolumn expansion would involve the/the/82009 amount of 111/1 / amount of 1 least 2009 row 1 solution 1st( 1st 28446 row expansion). Which Quizor7 or column expansion would involve 18 least work? following determinant by using w the 1. E1. Evaluate the following determinant by usingthe Laplace ExpansionMethod valuate ork? the Laplace Expansion Method ( 1st ( 1st row expansion). Which row or columnexpansion would involve the least amount ofof row expansion). Which row or column expansion would involve the least amount 2 21 1 3 3 116 6 66 111 1 work? work? 21 1 1 11 16 62( 2( 1 1)1 1 1) 1 ( 1)( 1)( 1 1)1 2 3( 3(1 1)1 3 1) 3 ( 1) 220 0 000 0 002 2 2 30 2 0 1 021 23 0 1 11 66 11 1 1 1 16 6 1)( 1 1 2 3( 1)1 3 1 1 1 62( 2( 1 1) 16 1) 1 ( (1)( 1)1) 2 3( 1)1 3 20 2 2[1(0)2 2(6)] [1(0) 6(0)]00 3[1(2) 1(0)]2 2[1(0) 0 2(6)] [1(0) 006(0)] 3[1(2)0 1(0)] 0020 20 3(2) 2( 2( 12) 0 0 3(2) 6(0)] 3[1(2) 1(0)] 12) 2[1(0) 2(6)] [1(0) 2[1(0) 2(6)] [1(0) 6(0)] 3[1(2) 1(0)] 18 18 3(2) 2( 2( 12) 0 0 3(2) 12) Expansion by 3rd row would entail least amount of work to to two 0 entries. Expansion by 8 rd row would entail least amount of work dueduethe the two 0 entries. 13 18 Expansion by 3 row would entail least amount of work due to the two 0 entries. 2. Evaluate by using1Row-reduction1to 1 angular form tri 1 10 0 0 0 1 1 10 0 0 0 2. Evaluate1by using 0 -reduction1to triangular form Row 0 1 0 01 1 10 01 1 00 , Row Row 100 1100 1 , Row 4 4 Row 1 , 1 , 0 1 0 0 10 1 2 1 20 11 0 0 0 1 1 2 10 1 1 0111 01 2 0 1 0 0 0 11 1 0 1 10110 1 0 1 1 00 0 1 01 102 0 1 1, Row 4 Row 1 , 1 0 2 , Row 4 Row 1 , 0 11 0 2 1 1 0 0 0 1 0 2 1 1 1 1 1 00 0 1 1 0 1 0 1 0 0011 0 1 1 11 0 00 0 1 0 1 , Row Row , Row 3 3 Row 2, 2, 0 30 1 0 0 0 0 1 1 31 1 01 , Row 3 Row 2, 0 0 0 0 1 01 0 111 1 10 3 , Row 3 Row 2, 0 0 0311 1 0 1 1 00 0 1 1 0 001 1 1 1 0 1 1 11 0 00 01 0 , Row 1 Row riangular form attained, , Row 4 4 Row 3, t3, triangular form attained, 13 3 01 03 1 0 0 0 0 1 1301 , 1 Row 4 Row 3, triangular form attained, 3 0 0 3 14 0 0 0 1 00 004 / 3 / 3 10 1 , Row 4 Row 3, triangular form attained, 06 M A T H 171000 28440 3 0 14 / 3 Q u i z 13 sol u t ion 8 11/23/ 2009 0 4 4 ) , determinant equal to product of diagonal entries of triangular form 0 (1)(1)( 3)(4/ ) ,3determinant equal to product of diagonal entries of triangular form 0 (1)(1)( 43)( 0 3 1. For the matrices3)A dand B given belowproduct ofinverse, ientries of ,triangular form (1)(1)( 3)( , eterminant equal to , find the diagonal f it exists by (i) R.Tam R.Tam (ii) R.Tam 2valuate by by using would entail to toamount form . Evaluate using Row-reduction least triangular form 2. EExpansion by 3rd rowRow-reductiontriangular of work due to the two 0 entries. rd R.Tam 3 44 Row 3)( ) , d method 4 (1)(1)( reduction eterminant equal to product of diagonal entries of triangular form 4 3 Adjoint method 4 1 3 1 1 4 2 0 det A det 3 1 3 1 1 1 0 2 0 2 ( 1) 23 A= 1 3 3 3 1 1 2 (3 3) 0 A has no inverse 1 0 1 1 0 det B 1( 1) 32 B= 1 1 1 1 0 [0 ( 1)] 1 1. For the matrices A and B given below, find the inverse, if it exists, by (i) Row reduction method M A T H 17100 28446 Q u i z 18 sol u t ion 11/23/ 2009 (ii) Adjoint method 1. For the matrices A and B given below, find the inverse, if it exists, by (i) A= Row reduction method 1 1 2 det A 1 3 4 3 1 3 3 det 1 3 1 1 1 1 1 1 1 0 2 0 0 2 0 2 ( 1) 23 det 2 ( 1) 23 3 3 1 1 2 (3 3) 0 (ii) 13 0 1 1 0 det B B = 1 1 0 A 0 as 1 inverse h no 0 1 exists B1 01 = 110 3 10 Adjoint method 1 no inverse 34 A has det A A= 1 1 2 3 3 1 1 2 (3 3) 0 1( 1) 32 1 1 0 [0 ( 1)] 1 (i) B Row reduction method: 010 det B 1( 1) 32 1 1 1 0 [0 ( 1)] 1 101100 1011 1 B exists 110010 0111 [ B | I ] (i) Row reduction0method: 010 01 0100 101100 1011 1 0 11 10 MATH 17100 28446 0 0 Quiz 19 solution 0 [ B | I ] 110010 0111 0 1 0 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0 1 0 1. A 10 0 For the matrix 0 0 010 00 1 1 0 0 1 1 1 1 1 1 1 0 01 10 00 1 0 0 0 0 1 00 0 1 10 00 01 11 1 11 1 [I | B ] 0 1 1 11/30/ 2009 1 0 1 1 314 01 0 2 6 , find 1 10 0 B 00 0 1 005 10 01 0 1 0 1 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 1 1 [I | B 1 0 1 1 1 (i) The eigenvalues 0 11 (ii) Adjoint method: (ii) B 1The 0 0 corresponding eigenvectors 1 1 1 1 Characteristic equation, (0 1) Cofactor matrix, C = 0 0 (0 0 0) 0 1 (1 0 0) 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 10 Adjoint method: 01 1 3 4 [0 ( 1)] det( A I ) det 0 2 6 (3 )(2 )(5 00 (0 0 ) 10 0 0 5 (0 1) 00 (1 0 ) Cofactor matrix, C = (ii) )0 0 1 1 3, 2, 5 0 1 [0 ( 1)] 1 0 For 1 3, suppose corresponding eigenvector is v (1) 33 1 0 23 0 0 4 6 53 0 0 0 14 16 0 2 010 001 000 v1(1) is a free variable ( (1) v21) and v3 are basic variables and ...
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## This note was uploaded on 11/04/2010 for the course MATH 17100 taught by Professor Kitchens during the Spring '10 term at IUPUI.

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