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quiz4_sol - Math 33A Section 3 Fall 2009 Quiz#4 Solutions 1...

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Math 33A, Section 3, Fall 2009 Quiz #4 Solutions 1. fl fl fl fl 2 3 4 6 fl fl fl fl = 2 · 6 - 3 · 4 = 12 - 12 = 0 2. fl fl fl fl - 9 4 - 2 1 fl fl fl fl = ( - 9) · 1 - 4 · ( - 2) = - 9 + 8 = - 1 3. There is only one non-zero pattern, and it has 3 inversions. fl fl fl fl fl fl 0 0 2 0 3 0 5 0 0 fl fl fl fl fl fl = - 2 · 3 · 5 = - 30 4. We subtract 2 times row I from row II, and 3 times row I from row III. fl fl fl fl fl fl 1 1 1 2 3 4 3 3 6 fl fl fl fl fl fl = fl fl fl fl fl fl 1 1 1 0 1 2 0 0 3 fl fl fl fl fl fl = 1 · 1 · 3 = 3 5. We first factor out 1 3 and 1 5 from rows I and III respectively, and then realize that the obtained determinant has two equal rows. fl fl fl fl fl fl 2 3 7 3 5 7 8 9 2 5 7 5 3 fl fl fl fl fl fl = 1 3 · 1 5 · fl fl fl fl fl fl 2 7 15 7 8 9 2 7 15 fl fl fl fl fl fl = 0 6. If we subtract row I from row III, we get precisely row II. fl fl fl fl fl fl fl fl 3 2 4 1 2 3 1 4 5 5 5 5 6 7 8 9 fl fl fl fl fl fl fl fl = fl fl fl fl fl fl fl fl 3 2 4 1 2 3 1 4 2 3 1 4 6 7 8 9 fl fl fl fl fl fl fl fl = 0 7. There is only one non-zero pattern, and it has 6 inversions. fl fl fl fl fl fl fl fl fl fl 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 fl fl fl fl fl fl fl fl fl fl = 1 · 1 · 1 · 1 · 1 · 1 = 1 1
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8. The matrix is lower-triangular. fl fl fl fl fl fl fl fl fl fl 1 0 0 0 0 2 - 1 0 0 0 3 - 2 1 0 0 4 - 3 2 - 1 0 5 - 4 3 - 2 1 fl fl fl fl fl fl fl fl fl fl = 1 ·
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