Inverse Matrices | 逆矩阵

📚 Inverse Matrices | 逆矩阵

In mathematics, the inverse of a matrix plays the same conceptual role as the reciprocal of a number. When a matrix is multiplied by its inverse, the product is the identity matrix, which acts like ‘1’ in matrix algebra. This idea is fundamental in solving simultaneous equations, in describing geometric transformations, and in many branches of applied mathematics.

在数学中，矩阵的逆与一个数的倒数起着类似的作用。当一个矩阵乘以其逆矩阵时，结果是单位矩阵，单位矩阵在矩阵代数中相当于’1’。这一概念在解联立方程组、描述几何变换以及许多应用数学分支中都是基础性的。

1. Square Matrices and Notation | 方阵与记号

A matrix is a rectangular array of numbers. An inverse can only be considered for a square matrix, one with the same number of rows and columns. If A is an n × n matrix, its inverse is usually written A⁻¹. The size of the matrix is called its order.

矩阵是一个矩形数阵。只有方阵才能考虑逆矩阵，即行数和列数相同的矩阵。若 A 是 n × n 矩阵，其逆矩阵通常记为 A⁻¹。矩阵的大小称为阶数。

For a 2 × 2 matrix, we often write A = [a b; c d], where the semicolon separates rows. Similarly, a 3 × 3 matrix has nine entries arranged in three rows and three columns.

对于 2 × 2 矩阵，我们常写成 A = [a b; c d]，其中分号分隔行。类似地，3 × 3 矩阵有九个元素，排列成三行三列。

2. The Identity Matrix | 单位矩阵

The identity matrix Iₙ is the square matrix with 1s on the leading diagonal and 0s elsewhere. For example, I₂ = [1 0; 0 1] and I₃ = [1 0 0; 0 1 0; 0 0 1]. It is the multiplicative identity: for any matrix A of the same order, A I = I A = A.

单位矩阵 Iₙ 是主对角线元素为 1、其余元素为 0 的方阵。例如，I₂ = [1 0; 0 1]，I₃ = [1 0 0; 0 1 0; 0 0 1]。它是乘法单位元：对任意同阶矩阵 A，有 A I = I A = A。

The identity matrix is the key to defining an inverse. If AB = I and BA = I, then B is the inverse of A.

单位矩阵是定义逆矩阵的关键。若有 AB = I 且 BA = I，则 B 是 A 的逆矩阵。

3. Definition of an Inverse Matrix | 逆矩阵的定义

Let A be a square matrix. If there exists a square matrix B such that AB = BA = I, then B is called the inverse of A. We write B = A⁻¹. Not every square matrix has an inverse; those that do are called invertible or non-singular.

设 A 为方阵。若存在方阵 B 使得 AB = BA = I，则称 B 为 A 的逆矩阵，记作 B = A⁻¹。并非每个方阵都有逆矩阵；有逆矩阵的矩阵称为可逆矩阵或非奇异矩阵。

The product of a matrix and its inverse is always the identity matrix, no matter the order of multiplication: A⁻¹ A = A A⁻¹ = I.

矩阵与其逆矩阵的乘积总是单位矩阵，无论相乘的顺序如何：A⁻¹ A = A A⁻¹ = I。

4. Determinant and Singular Matrices | 行列式与奇异矩阵

The determinant of a square matrix, written det(A) or |A|, is a scalar that encodes important properties. For the 2 × 2 matrix A = [a b; c d], det(A) = ad – bc.

方阵的行列式，记作 det(A) 或 |A|，是一个标量，包含许多重要性质。对于 2 × 2 矩阵 A = [a b; c d]，det(A) = ad – bc。

If det(A) = 0, the matrix has no inverse; it is called singular. If det(A) ≠ 0, the matrix is non-singular and its inverse exists.

若 det(A) = 0，则矩阵没有逆矩阵，称为奇异矩阵。若 det(A) ≠ 0，则矩阵非奇异，其逆矩阵存在。

Geometrically, a singular matrix maps the plane onto a line or a point, so the transformation cannot be undone. A non-zero determinant means the area scale factor of the transformation is non-zero, so the mapping is one-to-one.

几何上，奇异矩阵把平面映射到一条线或一个点，因此这种变换无法还原。非零行列式表示变换的面积比例因子非零，映射是一一对应的。

5. Finding the Inverse of a 2 × 2 Matrix | 求 2 × 2 矩阵的逆

For a 2 × 2 matrix A = [a b; c d], the inverse is given by:

A⁻¹ = 1 / (ad – bc) × [d -b; -c a]

The number ad – bc is the determinant. We swap the entries on the main diagonal, change the signs of the off-diagonal entries, and divide by the determinant.

数 ad – bc 就是行列式。我们将主对角线上的元素对调，改变副对角线上元素的符号，再除以行列式。

Example: if A = [2 1; 5 3], then det(A) = 2 × 3 – 1 × 5 = 1. Thus A⁻¹ = [3 -1; -5 2]. Check: [2 1; 5 3][3 -1; -5 2] = [1 0; 0 1].

例如：若 A = [2 1; 5 3]，则 det(A) = 2 × 3 – 1 × 5 = 1。因此 A⁻¹ = [3 -1; -5 2]。验证：[2 1; 5 3][3 -1; -5 2] = [1 0; 0 1]。

6. Finding the Inverse of a 3 × 3 Matrix | 求 3 × 3 矩阵的逆

For a 3 × 3 matrix, the inverse is found using the adjugate method. The steps are:

对于 3 × 3 矩阵，常用的方法是伴随矩阵法。步骤如下：

Calculate the matrix of minors for each entry.

计算每个元素的余子式矩阵。

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Inverse Matrices | 逆矩阵