assign9_practice_soln

assign9_practice_soln - Math 136 Practice Problems # 9 213...

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Unformatted text preview: Math 136 Practice Problems # 9 213 1. Let A = 3 a −1. Assuming that A is invertible, find A−1 by the cofactor method. −2 1 4 4a + 1 −10 3 + 2a 14 −4 , so Solution: We have cof(A) = −1 −1 − 3a 11 2a − 3 4a + 1 −1 −1 − 3a 11 (cof(A))T = −10 14 3 + 2 a −4 2 a − 3 Moreover, by definition of the determinant we have the determinant of A is the dot product of the i-th row of A with the i-th column of (cof(A))T . Hence we get Hence, A−1 det A = 3(−1) + a(14) + (−1)(−4) = 1 + 14a 4a + 1 −1 −1 − 3a 1 1 11 . = det A (cof(A))T = 1+14a −10 14 3 + 2a −4 2a − 3 2. Use Cramer’s Rule to solve the following systems. a) 3x1 − x2 = 2 4x1 + 7x2 = 5 Solution: The coefficient matrix is A = 3 −1 , so det A = 21 + 4 = 25. Hence, 47 1 25 1 x2 = 25 x1 = Thus, the solution is x = 19 2 −1 = 57 25 7 32 = 45 25 19/25 . 7/25 1 2 b) 2x1 + 3x2 + x3 = 1 x1 + x2 − x3 = −1 −2x1 + 2 x3 = 1 231 Solution: The coefficient matrix is A = 1 1 −1, so det A = 6. Hence, −2 0 2 131 1 4 −1 1 −1 = x1 = 6102 6 x2 = 2 1 1 1 −3 1 −1 −1 = 6 −2 1 6 2 x3 = 231 7 1 1 1 −1 = 6 −2 0 1 6 2/3 Hence, the solution is x = −1/2. 7/6 c) 2x1 + x2 = 1 3x1 + 7x2 = −2 Solution: The coefficient matrix is A = 21 , so det A = 14 − 3 = 11. Hence, 37 1 11 1 x1 = 11 x1 = Thus, the solution is x = 9/11 . −7/11 9 11 = −2 7 11 7 21 = 3 −2 11 3 d) 5x1 + x2 − x3 = 4 9x1 + x2 − x3 = 1 x1 − x2 + 5 x3 = 2 5 1 −1 Solution: The coefficient matrix is A = 9 1 −1, so det A = −16. Hence, 1 −1 5 4 1 −1 1 12 1 1 −1 = x1 = −16 2 −1 5 −16 x2 = 5 4 −1 1 −166 9 1 −1 = −16 1 2 5 −16 x3 = 514 −42 1 9 1 1= −16 1 −1 2 −16 −3/4 Thus, the solution is x = 83/7 . 21/8 x1 1 1a 2 0 1 b , x = x2 and b = −1. Assuming that A is invertible, 3. Let A = 2 x3 1 c −1 use Cramer’s Rule to find the value of x2 in the solution of the equation Ax = b. 1a 2 Solution: We find that det A = 0 1 b = 1 c −1 11 2 11 1 1 So, x2 = det A 0 −1 b = −3−b(c−a) 0 −1 1 2 −1 01 1 a 2 0 1 b = −3 − b(c − a). 0 c−a 3 2 3−b b = −3−b(c−a) . −3 ...
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This note was uploaded on 07/13/2011 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

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assign9_practice_soln - Math 136 Practice Problems # 9 213...

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