-mtrxbeamer2 - Chapter 2 Determinants Matrix Theory Leonor...

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Chapter 2 Determinants Matrix Theory Leonor Aquino-Ruivivar Mathematics Department De La Salle University-Manila (L.A. Ruivivar) Lectures in Linear Algebra 1 / 36
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Permutations Definition: Let S = { 1 , 2 , . . . , n } be the set of the first n positive integers. A rearrangement σ = j 1 j 2 . . . j n of the elements of S is called a permutation of the set S . Example: If n = 3, then S = { 1 , 2 , 3 } . There are six possible rearrangements of the elements of the set S . These are: 123, 132, 213, 231, 312 and 321. Each of these rearrangements is a permutation of S . The first permutation has j 1 = 1 , j 2 = 2 and j 3 = 3. Remark: Since S has n elements, there are n ! permutations of the set S . (L.A. Ruivivar) Lectures in Linear Algebra 2 / 36
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Permutations Definition: Let S = { 1 , 2 , . . . , n } be the set of the first n positive integers. A rearrangement σ = j 1 j 2 . . . j n of the elements of S is called a permutation of the set S . Example: If n = 3, then S = { 1 , 2 , 3 } . There are six possible rearrangements of the elements of the set S . These are: 123, 132, 213, 231, 312 and 321. Each of these rearrangements is a permutation of S . The first permutation has j 1 = 1 , j 2 = 2 and j 3 = 3. Remark: Since S has n elements, there are n ! permutations of the set S . (L.A. Ruivivar) Lectures in Linear Algebra 2 / 36
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Permutations Definition: Let S = { 1 , 2 , . . . , n } be the set of the first n positive integers. A rearrangement σ = j 1 j 2 . . . j n of the elements of S is called a permutation of the set S . Example: If n = 3, then S = { 1 , 2 , 3 } . There are six possible rearrangements of the elements of the set S . These are: 123, 132, 213, 231, 312 and 321. Each of these rearrangements is a permutation of S . The first permutation has j 1 = 1 , j 2 = 2 and j 3 = 3. Remark: Since S has n elements, there are n ! permutations of the set S . (L.A. Ruivivar) Lectures in Linear Algebra 2 / 36
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Permutations Definition: Let S = { 1 , 2 , . . . , n } be the set of the first n positive integers. A rearrangement σ = j 1 j 2 . . . j n of the elements of S is called a permutation of the set S . Example: If n = 3, then S = { 1 , 2 , 3 } . There are six possible rearrangements of the elements of the set S . These are: 123, 132, 213, 231, 312 and 321. Each of these rearrangements is a permutation of S . The first permutation has j 1 = 1 , j 2 = 2 and j 3 = 3. Remark: Since S has n elements, there are n ! permutations of the set S . (L.A. Ruivivar) Lectures in Linear Algebra 2 / 36
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Odd and Even Permutations Definition: A permutation σ = j 1 j 2 . . . j i · · · j k · · · j n is said to have an inversion if a larger number j i precedes a smaller number j k . If the total number of inversions in σ is odd (even) , then σ is called an odd (even) permutation . Example: Let n = 3 and let σ = 312. Then σ contains the inversions 31 and 32. Since the number of inversions is even, σ is an even permutation. The following table shows the permutations for n = 3, the corresponding inversions and their classification as odd or even permutations. Permutation Inversions Classification 123 none even 132 32 odd 213 21 odd 231 21 and 31 even 312 31 and 32 even 321 32 , 31 and 21 odd (L.A. Ruivivar) Lectures in Linear Algebra 3 / 36
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Odd and Even Permutations Definition: A permutation σ = j 1 j 2 . . . j i · · · j k · · · j n is said to have an inversion if a larger number j i
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