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Determinants ( 行列式 )

Determinants ( 行列式 ). ● Permutations ( 排列 ) A Permutation of order n : 1, 2, …, n 任意排列 . n! different permutations. ● Example: P1=1,3,2 is a odd permutation of order 3. P1(1)=1, P1(2)=3. <Definition> Determinant Let A = [ a ij ] be an n x n matrix.

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Determinants ( 行列式 )

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  1. Determinants (行列式) ● Permutations (排列) A Permutation of order n : 1, 2, …, n 任意排列. n! different permutations. ● Example: P1=1,3,2 is a odd permutation of order 3. P1(1)=1, P1(2)=3 <Definition> Determinant Let A = [ aij ] be an n x n matrix. The determinant of A = det(A)= |A| is the sum of all products take over all permutations p on 1, 2, …, n.

  2. ● 2 x 2 Matrix All permutations of order n (+) (-) ● 3 x 3 Matrix : 6 permutations … …

  3. Some Properties of Determinant ● If A has a zero row (column), then |A|=0. ● Let B be formed from A by multiplying row k by a scalar α , Then |B| = α |A|. ● Let B be formed from A by interchange two rows (columns), Then |B| = - |A|. ● If two rows of A are the same, then |A|=0. Proof: ● If for some scalar α, row k of A is α times row I, then |A|=0. Proof: ● | A | = | At | 亦即，前述對 row的特性結果，對column也成立。 ● Let A and B be n x n matrices. Then |AB| = |A| |B|. Proof later!

  4. <Theorem> Where Example:

  5. ● Let B be formed from A by adding γtime the row i to row k , then |B|=|A| . Proof:

  6. ● 對 A 做 Elementary Row Operations 時，|A| 的變化： Type I : Interexchange two rows of A  |A| → -|A| Type II : Multiply a row of A by a nonzero constant α  |A| → α|A| Type III : Add a scalar multiple of one row to another one  |A| → |A| ● 可利用 Type III 使某一 column (row) 只剩一項不為 0, 以簡化計算。 Example：

  7. <Definition> Minor (次階, 降階) If A is an n x n matrix, the minor of aij is denoted Mij, and is the determinant of the (n-1) x (n-1) matrix obtained by deleting row i and column j of A. ● Co-factor : The number (-1)i+j Mij is called the cofactor of aij ● The cofactor expression of |A| :

  8. <Theorem> Cofactor Expression by Row If A is n x n matrix, then for any integer i with <Theorem> Cofactor Expression by Column If A is n x n matrix, then for any integer j with Example：

  9. Determinants of Triangular Matrices a11,a12, … ,ann : main diagonal of a square matrix A A is “lower triangular” A is “upper triangular” Determinants of a Triangular Matrix

  10. ● 對 A 做 Elementary Row Operations 時，|A| 的變化： Type I : Interexchange two rows of A  |A| → - |A| Type II : Multiply a row of A by a nonzero constantα  |A| → α|A| Type III : Add a scalar multiple of one row to another one  |A| → |A| 總效果： When  When A is nonsingular, AR is also nonsingular  AR = In  Ω = A-1  Theorem (A : n x n matrix)

  11. <Theorem> • Let A be an n x n matrix. Then A is nonsingular if and only if • Proof: • Assume • A is nonsingular  AR = In  AR has no zero row A has inverse (A-1 = Ω)  A is nonsingular <Theorem> If A and B are n x n, and either is singular. Then AB and BA are both singular. Proof:

  12. <Theorem> Formula of Inverse Let A be an n x n nonsingular matrix. Define an n x n matrix B by putting Then i, j element of A-1 is the cofactor of aj,i divided by |A| Proof: Example:

  13. Cramer’s Rule <Theorem> Cramer’s Rule Let A be a nonsingular n x n matrix of numbers. The unique solution of AX=B is is the matrix obtained from A by replacing column k of A by B. Proof:

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