Explanation Procedure used: 1. Form an augmented matrix [ A | I ] where a matrix A is placed along with a same dimension identity matrix I separated by a line. 2. Apply row operations on [ A | I ] in order to obtain another augmented matrix [ I | B ] where I , is again an identity matrix and B is an inverse of A. 3. If the reduction process yields a row of zeros in the left half of the augmented matrix, then the inverse of A does not exist. Calculation: Let x , y and z be the percent of venture capital from business loans, auto loans and home loans respectively. Since the total capital in the year 2001 is 483.94 million, the first equation is 532 x + 58 y + 682 z = 483.94 . Since the total capital in the year 2002 is 503.28 million, the second equation is 562 x + 62 y + 695 z = 503.28 . Since the total capital in the year 2003 is 521.33 million, the second equation is 578 x + 69 y + 722 z = 521.33 . Thus, the system of equations becomes, { 532 x + 58 y + 682 z = 483.94 562 x + 62 y + 695 z = 503.28 578 x + 69 y + 722 z = 521.33 Convert the system of equation into a matrix notation. [ 532 58 682 562 62 695 578 69 722 ] [ x y z ] = [ 483.94 503.28 521.33 ] Take A = [ 532 58 682 562 62 695 578 69 722 ] and X = [ x y z ] and transform [ 532 58 682 562 62 695 578 69 722 ] [ x y z ] = [ 483.94 503.28 521.33 ] as follows. A − 1 A X = A − 1 [ 483.94 503.28 521.33 ] ( Multiply with A − 1 on RHS of both sides ) I X = A − 1 [ 483.94 503.28 521.33 ] ( ∵ A − 1 A = I ) X = A − 1 [ 483.94 503.28 521.33 ] Obtain the augmented matrix as follows. [ 532 58 682 1 0 0 562 62 695 0 1 0 578 69 722 0 0 1 ] Apply the row operation 281 R 1 − 266 R 2 → R 2 and 289 R 1 − 266 R 3 → R 3 and reduce the matrix. [ 532 58 682 1 0 0 0 − 194 6772 281 − 266 0 0 − 1592 5046 289 0 − 266 ] Apply the row operation 97 R 1 + 29 R 2 → R 1 and 796 R 2 − 97 R 3 → R 3 and reduce the matrix. [ 51 , 604 0 262 , 542 8 , 246 − 7 , 714 0 0 − 194 6772 281 − 266 0 0 0 4 , 901 , 050 195 , 643 − 211 , 736 25 , 802 ] Apply the row operation 18 , 425 R 1 − 987 R 3 → R 1 and 2,450,525 R 2 − 3 , 386 R 3 → R 2 and reduce the matrix. [ 950 , 803 , 700 0 0 − 41 , 167 , 091 66 , 852 , 982 − 25 , 466 , 574 0 − 475 , 401 , 850 0 26 , 150 , 327 65 , 098 , 446 − 87 , 365 , 572 0 0 4 , 901 , 050 195 , 643 − 211 , 736 25 , 802 ] Apply the row operations 1 950 , 803 , 700 R 1 → R 1 , − 1 475 , 401 , 850 R 2 → R 2 and 1 4 , 901 , 050 R 3 → R 3 and reduce the matrix

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Chapter 7.4, Problem 33E

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Chapter 7.4, Problem 32E

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Chapter 7 Solutions

EBK COLLEGE ALGEBRA IN CONTEXT

Ch. 7.1 - In Exercises 14, solve the systems of equations....Ch. 7.1 - Prob. 2E Ch. 7.1 - Prob. 3E Ch. 7.1 - Prob. 4E Ch. 7.1 - Prob. 5E Ch. 7.1 - Prob. 6E Ch. 7.1 - Prob. 7E Ch. 7.1 - Prob. 8E Ch. 7.1 - Prob. 9E Ch. 7.1 - Prob. 10E

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