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fm241sol1 - M241 Matrix Algebra Homework Assignment 1...

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M241: Matrix Algebra Homework Assignment 1 Answers Graded Problems: Section 1.2 Problem 4; Section 1.3 Problems 10 and 12; Section 1.4 Problems 2 and 38; Section 1.5 Problem 4. Section 1.2. Problem 4. No solution; zero right sides produce 3 lines through origin; right sides 1, 1, 0 give x = 1 and y = 0; right sides 2, -1, 1 give x = 0 and y = 1; any combination is solvable. Problem 8. eq(1)-eq(2)+eq(3) gives 0=1; change 0 to -1; (3 , - 1 , 0) is a solution. Problem 22. An example is ( a, b ) = (3 , 6) and ( c, d ) = (1 , 2). If ( a, b ) = k ( c, d ) and abcd 6 = 0, then k = a/c = b/d . Now a/c = b/d implies that ad = bc and thus a/b = c/d . Thus ( a, c ) = l ( b, d ), where l = a/b = c/d . Q.e.d. Section 1.3. Problem 1. Multiply by l = 10 / 2 = 5 and subtract to find 2 x + 3 y = 1 and - 6 y = 6. The pivots are 2 and - 6. Problem 4. Subtract l = c/a times equation 1. The new second pivot multiplying y is d - cb a = ad - bc a . Then y = ag - cf ad - bc . Problem 8. If k = 3 elimination must fail: no solution. If k = - 3, elimination gives 0 = 0 in equation 2: infinitely many solutions. If k = 0, a row exchange is needed: one solution.
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