math102HW6 - Math 102 Homework 6(selected problems David...

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Math 102 - Homework 6 (selected problems) David Lipshutz Problem 1. (Strang, 4.4: #14) Use Cramer’s Rule to solve for y . ( a ) ax + by = 1 cx + dy = 0 ( b ) ax + by + cz = 1 dx + ey + fz = 0 gx + hy + iz = 0 Proof. (a) A = " a b c d # b = " 1 0 # B 2 = " a 1 c 0 # y = det B 2 det A = - c ad - bc (b) A = a b c d e f g h i b = 1 0 0 B 2 = a 1 c d 0 f g 0 i y = det B 2 det A = - di - fg det A Problem 2. (Strang, 4.4: #18) Find A - 1 from the cofactor formula C T / det A . Use symmetry in part (b): (a) A = 1 2 0 0 3 0 0 4 1 (b) A = 2 - 1 0 - 1 2 - 1 0 - 1 2 Proof. (a) C = 3 0 0 - 2 1 - 4 0 0 3 A - 1 = C T det A = 1 3 3 - 2 0 0 1 0 0 - 4 3 (b) C = 3 2 1 2 4 2 1 2 3 A - 1 = C T det A = 1 4 3 2 1 2 4 2 1 2 3 1
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Problem 3. (Strang, 4.4: #36) The triangle with corners (0 , 0) , (1 , 0) , (0 , 1) has area 1 2 . The pyramid with four corners (0 , 0 , 0) , (1 , 0 , 0) , (0 , 1 , 0) , (0 , 0 , 1) has what volume? The pyramid with five corners at (0 , 0 , 0 , 0) and the rows of I has what volume?
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