Inverse Matrix

Table of Contents

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Definition

Inverse matrix is a fundamental concept in linear algebra.

When multiplied with the original matrix, produces the identity matrix.

For a square matrix $A$

$A$ inverse is denoted as $A^{-1}$
And satisfies the following relationship

$$ A × A^{-1} = I $$
- where $I$ is the identity matrix.

Formula

$$ A^{-1} = \frac{1}{det(A)} \cdot adj(A) $$

Find the inverse matrix of $A$.
- $det(A)$: determinant
  - Represents the characteristic value of matrix $A$
- $adj(A)$: adjoint matrix
  - The transpose of the cofactor matrix

Important Points

Inverse matrix exists only when ($det(A) \neq 0$)
if ($det(A) = 0$), the matrix is called a singular matrix and has no inverse </aside>

Example

Let's find the inverse matrix of $4{×}4$ matrix A:

$$ A = \begin{bmatrix} 2 & -1 & 3 & 0\\ 1 & 4 & -2 & 5\\ 0 & 2 & 1 & 3\\ 3 & -3 & 2 & 1 \end{bmatrix} $$
First, we find $M₁₁$.

$$ M_{11} = \det\begin{bmatrix} 4 & -2 & 5\\ 2 & 1 & 3\\ -3 & 2 & 1 \end{bmatrix} = 37 $$
We get the minor matrix $M$.

$$ M = \begin{bmatrix} 37 & -38 & 17 & 31\\ -28 & 17 & 13 & -7\\ -45 & 18 & 24 & -33\\ 5 & -31 & 7 & 23 \end{bmatrix} $$
We can get the cofactor matrix $C_A$.

$$ C_A = \begin{bmatrix} 37 & 38 & 17 & -31\\ 28 & 17 & -13 & -7\\ -45 & -18 & 24 & 33\\ -5 & -31 & -7 & 23 \end{bmatrix} $$

Adjoint matrix is the transpose of the cofactor matrix.

$$ \text{adj}(A) = C_A^T = \begin{bmatrix} 37 & 28 & -45 & -5\\ 38 & 17 & -18 & -31\\ 17 & -13 & 24 & -7\\ -31 & -7 & 33 & 23 \end{bmatrix} $$

Finally, we calculate the determinant of $A$.

$$ det(A) = 87 $$
Therefore, the inverse matrix =

$$ A^{-1} = \frac{1}{87}\begin{bmatrix} 37 & 28 & -45 & -5\\ 38 & 17 & -18 & -31\\ 17 & -13 & 24 & -7\\ -31 & -7 & 33 & 23 \end{bmatrix} $$

$$ A^{-1} = \begin{bmatrix} 0.42529 & 0.32184 & -0.51724 & -0.05747\\ 0.43678 & 0.19540 & -0.20690 & -0.35632\\ 0.19540 & -0.14943 & 0.27586 & -0.08046\\ -0.35632 & -0.08046 & 0.37931 & 0.26437 \end{bmatrix} $$

Properties

Inverse of Inverse

Applying the inverse twice returns the original matrix.

$$ (A^{-1})^{-1} = A $$

Inverse of Producet

The inverse of a product reverses the order of multiplication.

$$ (AB)^{-1} = B^{-1}A^{-1} $$

Transpose of Inverse

The transpose of inverse equals the inverse of the transpose.

$$ (A^T)^{-1} = (A^{-1})^T $$

Determinant Relationship

$$ det(A^{-1}) = \frac{1}{det(A)} $$

Applications

Why Use Inverse Matrix?

Purpose	Description
행렬 방정식 해결	선형 방정식 시스템의 해를 구하는 데 사용
변환 역변환	좌표계 변환이나 기하학적 변환을 되돌리는 데 활용
최적화 문제	최적의 해를 찾기 위한 수학적 연산에 필요
데이터 분석	회귀 분석과 같은 통계적 방법에서 중요한 역할

Computer Graphics

Inverse Matrix는 컴퓨터 그래픽스에서 변환(Transformation)을 뒤집거나 복원할 때 필요합니다.

Camera View Transformation
Object Transformation Undo

Practice with Python

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Following the previous code snippet, you can get an inverse matrix by using the functions below.

adj_A = adjoint(A)
det_A = np.linalg.det(A)

inverse_A = (1 /det_A) * adj_A

print(inverse_A)

Finally we can observe the result!

[[ 0.42528736  0.32183908 -0.51724138 -0.05747126]
 [ 0.43678161  0.1954023  -0.20689655 -0.35632184]
 [ 0.1954023  -0.14942529  0.27586207 -0.08045977]
 [-0.35632184 -0.08045977  0.37931034  0.26436782]]

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