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Chapter 3. Determinants and Inverses

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1. Determinant

2. Inverse Matrix

3. Cramer’s Rule

1. Determinant

⑴ Definition of a permutation

① Permutation: A permutation of S = {1, 2, ···, n} refers to a one-to-one correspondence function σ from S to S.

○ This function is simply written as (f(1), ···, f(n)).

○ Identity permutation: (1, 2, ···, n). That is, when f(i) = i.

② Transposition: A transformation that changes (f(1), ···, f(i), ···, f(j), ···, f(n)) to (f(1), ···, f(j), ···, f(i), ···, f(n))

○ In other words, a transposition swaps the images of two elements in the domain of a permutation.

○ Any permutation can be formed by multiplying finitely many transpositions from (1, ···, n).

③ Sign of a permutation

○ Odd permutation: A permutation that can form the identity permutation by multiplying an odd number of transpositions.

○ Even permutation: A permutation that can form the identity permutation by multiplying an even number of transpositions.

○ If a permutation σ is odd, define sgn(σ) = -1; if even, sgn(σ) = +1.

④ Parity of the symmetric group: A permutation cannot be both even and odd.

○ First assume f(0) = 0.

○ Inversion

○ For α ≤ x ≤ β, define M(k)_{α, β} as the number of x such that f(x) ＞ f(y), called an inversion.

○ Sum of inversions

○ Definitions

○ n₁: Number of n such that f(n) > f(a) and 0 < n < i.

○ n₂: Number of n such that f(a) > f(n) > f(b) and 0 < n < i.

○ n₃: Number of n such that f(b) > f(n) and 0 < n < i.

○ n₄: Number of n such that f(n) > f(a) and i < n < j.

○ n₅: Number of n such that f(a) > f(n) > f(b) and i < n < j.

○ n₆: Number of n such that f(b) > f(n) and 0 < n < i.

○ Case 1. When f(i) = a and f(j) = b (assuming f(a) ＞ f(b))

Figure 1. First case of permutation

○ Case 2. When f(i) = b and f(j) = a (assuming f(a) ＞ f(b))

Figure 2. Second case of permutation

○ Conclusion: Permutations have parity.

⑵ Definition of determinant: When [A]^j is the j-th column vector of matrix A,

① Property 1. det[I] = 1

② Property 2. Property related to the sign of permutations

○ 2-1. If in A = ([A]¹, ···, [A]ⁿ), [A]^h = [A]^k (1 ≤ h ≠ k ≤ n), then det(A) = 0

③ Property 3. det(A) = det([A]¹, ···, [A]ⁿ) is a multilinear function in the column vectors.

○ 3-1. If [A]ⁱ = 0 in A = ([A]¹, ···, [A]ⁿ), then det(A) = 0 (∵ 0 + 0 = 0)

○ 3-2. Useful operations on columns: Since the determinant remains the same for the transpose matrix, operations on rows are also allowed.

④ The unique function satisfying Property 1, Property 2, and Property 3 is as follows.

⑶ Computing determinants

① Determinant of a 2 × 2 matrix

② Determinant of a 3 × 3 matrix

③ Cofactor expansion: Computing determinants using minors

⑷ Applications of determinants

① Theorem 1. If the rank of an n × n matrix A is less than n, then det(A) = 0.

○ Reason: The determinant implies the existence of a unique solution via the inverse matrix.

② Theorem 2. Determinant of the transpose: If A is an n × n matrix, det(A^t) = det(A)

③ Theorem 3. If A and B are n × n matrices, det(AB) = det(A)det(B)

④ Theorem 4. For X ∈ ℳ_n, A ∈ ℳ_r, D ∈ ℳ_n-r, B ∈ ℳ_{r, n-r}, the following holds:

○ Theorem 4-1.

○ Theorem 4-2. For n × n matrices A, B, C, D, if ㅣDㅣ ≠ 0 and CD = DC

○ Theorem 4-3. For A ∈ ℳ_r, B ∈ ℳ_{r, n-r}, C ∈ ℳ_{n-r, r}, D ∈ ℳ_n-r, the following identity holds.

⑤ Theorem 5. When x and y are both n-dimensional column vectors, the following identity holds.

⑥ Theorem 6. Vandermonde matrix

○ Definition: A matrix whose columns consist of powers of certain numbers.

○ Determinant

○ The given determinant can be expressed as a polynomial: For example, regarding x_n, the polynomial is of degree ≤ n-1 in x_n.

○ Note the fact that subtracting one row vector from another does not change the determinant.

○ From the difference between the nth row and the 1st row, the determinant has (x_n - x₁) as a factor; similarly (x_n - x₂), ···, (x_n - x_n-1).

○ Thus, the polynomial is expressed as (x_n - x₁) ··· (x_n - x_n-1): Confirm the coefficient of x_n^n-1 is 1.

○ Without loss of generality, the same pattern holds for x_n-1, x_n-2, ···, x₁, proving the formula.

⑦ Theorem 7. Gramian (Gram) matrix

○ Definition: A symmetric matrix formed by inner products of vectors.

⑧ Theorem 8. Hankel matrix

○ Definition: A matrix in which all values along the anti-diagonal (bottom-left ↗ top-right) are equal.

⑸ Uses of determinants

① Use 1. Determinants represent area, volume, etc., transformed by a linear transformation.

Figure 3. Determinant and area of a 2-dimensional linear transformation

② Use 2. If given row vectors or column vectors are dependent, the determinant is 0; if independent, the determinant is non-zero (and the converse is also true).

○ Case 1. Dependent: By Property 2, the determinant becomes 0.

○ Case 2. Independent

○ Using Gaussian elimination to form an upper triangular matrix, none of the diagonal elements become 0.

○ Such an upper triangular matrix clearly has a non-zero determinant.

○ The determinant does not change during Gaussian elimination.

○ Therefore, a matrix with independent column or row vectors has a non-zero determinant.

③ Use 3. Existence of inverse matrices: A determinant of 0 is a necessary and sufficient condition for a matrix to have no inverse.

○ Because Gaussian elimination, which preserves the determinant, can automatically find the inverse.

④ Use 4. Equation of a line passing through (x₁, y₁)’, (x₂, y₂)’ in the x-y plane

⑤ Use 5. Cross product of v = (v₁, v₂, v₃) and w = (w₁, w₂, w₃)

⑥ Use 6. Wronskian matrix

⑦ Use 7. Jacobian matrix

⑧ Use 8. Hessian matrix

○ Hessian for a 2 × 2 matrix

○ Hessian for a 3 × 3 matrix

○ Hessian for an n × n matrix

⑨ Use 9. ► Hückel approximation and Hamiltonian (ref))

⑩ Use 10. Rotation and reflection transformations

○ Rotation transformation: For a matrix R representing T: x → y, if R^TR = RR^T = I (isometric), det(R) = 1

○ Reflection transformation: For a matrix R representing T: x → y, if R^TR = RR^T = I (isometric), det(R) = -1

○ Example applied to an object’s collision

⑪ Use 11. Three points are collinear if and only if the determinant of the matrix formed by their coordinate vectors is zero.

○ Example: Proof of Desargues’ Theorem

2. Inverse Matrix

⑴ Definition

⑵ Invertible matrix: A matrix with an existing inverse.

① Theorem 1. If A has an inverse, it is unique.

② Theorem 2. If A is invertible, A^-1 is also invertible.

③ Theorem 3. If A and B are invertible, AB is also invertible.

④ Theorem 4. If A is invertible, cA is also invertible.

⑤ Theorem 5. Inverse matrix and transpose matrix

⑥ Theorem 6. If there exists a natural number k such that det (A^k) ≠ 0 (A is 3 × 3), then A is invertible.

⑦ Theorem 7. If there exists a natural number k such that (I - A^k) = O (A is 3 × 3, I is the identity matrix), then A is invertible.

⑧ Theorem 8. If there exist v₁, v₂, ···, v_k such that Av₁ ∪ Av₂ ∪ ··· ∪ Av_k = ℝ³, then A is invertible.

○ Reason: It means rank (A) = 3.

⑨ The existence of three orthogonal vectors v₁, v₂, v₃ with Av_i ≠ O is not sufficient to regard A as invertible.

○ Reason: Because A may map vectors to a single non-zero vector.

⑶ Computing inverses 1. Finding inverses by transforming to triangular matrices

① Elementary row operations: Provide an algorithm for finding inverse matrices.

○ ⓐ Swapping row i and row j of A is equivalent to swapping row i and row j of C.

○ ⓑ Multiplying row i of A by c gives the same effect as multiplying row i of C by c.

○ ⓒ Adding c times row j of A to row i is equivalent to performing the same operation on C.

② Elementary matrix: A matrix that has the same effect as applying a single elementary row operation.

○ Type 1 elementary matrix: Represents the first elementary operation (ⓐ).

○ Type 2 elementary matrix: Represents the second elementary operation (ⓑ).

○ Type 3 elementary matrix: Represents the third elementary operation (ⓒ).

③ Theorem 1. All elementary matrices are invertible.

○ The inverse of a type 1 elementary matrix is itself.

○ Inverse of a type 2 elementary matrix

○ Inverse of a type 3 elementary matrix

④ Theorem 2. Any invertible matrix can be expressed as a product of elementary matrices.

⑤ Theorem 3. All triangular matrices are invertible.

○ Triangular matrix: Either upper triangular or lower triangular.

○ Any n × n triangular matrix A can be expressed as a product of elementary matrices.

○ First assume the given triangular matrix is upper triangular.

○ Step 1. From the identity matrix, multiply row 2 by a₁₂ and add it to row 1.

○ Step 2. After Step 1, multiply row 3 by a₁₃ and add to row 1, and multiply by a₂₃ and add to row 2.

○ Step 3. After Step 2, multiply row 4 by a₁₄ and add to row 1, multiply by a₂₄ and add to row 2, and multiply by a₃₄ and add to row 3.

○ Step 4. By repeating this method, the given upper triangular matrix can be expressed using only the identity matrix.

○ The same process works for lower triangular matrices.

○ Therefore, triangular matrices have inverses and their forms can be determined.

⑥ Theorem 1. Calculation of det($A$ + $B$) given an arbitrary matrix $A$ and a matrix $B$ with rank 1

⑷ Computing inverses 2. Adjoint matrix (Adjugate matrix)

① Minor: When A is an $n$ × $n$ matrix, removing row i and column j results in an $(n-1)$ × $(n-1)$ matrix.

② Cofactor a_ij

③ Laplace expansion

④ Adjoint matrix

⑤ Properties of adjoint matrix

⑸ Computing inverses 3. Programming code

① Using Python numpy

import numpy as np

# Define an example matrix A
A = np.array([
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9]
])

# Obtain inverse matrix using numpy.linalg.inv function
try:
    A_inv = np.linalg.inv(A)
    print("A_inv (NumPy):")
    print(A_inv)
except np.linalg.LinAlgError:
    print("There is no inverse matrix.")

② Using Python sympy

import sympy

# Define an example matrix A
A_sym = sympy.Matrix([
    [1, 2, 1],
    [0, 1, 2],
    [1, 2, 3]
])

# Obtain inverse matrix 
try:
    A_sym_inv = A_sym.inv()  # .inv() 메서드 사용
    print("A_inv (Sympy):")
    print(A_sym_inv)
except:
    print("There is no inverse matrix.")

3. Cramer’s Rule

⑴ Let A be an n × n matrix, x = (x₁, ···, x_n)’, b = (b₁, ···, b_n)’. The j-th element of the unique solution x₀ of Ax = b is:

⑵ Proof

Input: 2020.04.22 09:51

Revised: 2024.01.01 18:37