The Adjoint and Inverse Matrices | 伴随矩阵与逆矩阵

📚 The Adjoint and Inverse Matrices | 伴随矩阵与逆矩阵

In linear algebra, the inverse of a matrix is a central concept for solving systems of equations, analysing transformations, and understanding matrix structure. The adjoint (or adjugate) matrix provides an elegant algebraic route to constructing the inverse using determinants and cofactors. This article explores the definitions, properties and worked examples of adjoint and inverse matrices, tailored to the IB Mathematics curriculum.

在线性代数中，逆矩阵是解方程组、分析变换以及理解矩阵结构的核心概念。伴随矩阵（或称伴随矩阵）为利用行列式和余子式构造逆矩阵提供了一条优美的代数路径。本文探讨伴随矩阵和逆矩阵的定义、性质及计算范例，内容贴合 IB 数学课程。

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1. Introduction to Inverse Matrices | 逆矩阵简介

An inverse matrix of a square matrix A is a matrix A^-1 such that AA^-1 = A^-1A = I, where I is the identity matrix of the same order. Only square matrices can have inverses, and not all square matrices are invertible.

一个方阵 A 的逆矩阵是一个矩阵 A^-1，使得 AA^-1 = A^-1A = I，其中 I 是同阶单位矩阵。只有方阵才能有逆矩阵，并且并非所有方阵都可逆。

If an inverse exists, it is unique. The inverse reverses the transformation represented by the matrix, much like a reciprocal for numbers. In IB Mathematics, inverse matrices are used to solve linear systems, compute transformation inverses, and simplify matrix expressions.

如果逆矩阵存在，则它是唯一的。逆矩阵逆转了矩阵所表示的变换，就像数的倒数一样。在 IB 数学中，逆矩阵用于解线性方程组、计算变换的逆以及简化矩阵表达式。

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2. Determinants and Singular Matrices | 行列式与奇异矩阵

The determinant of a square matrix, denoted det(A) or |A|, is a scalar value that determines whether the matrix is invertible. A matrix with a non-zero determinant is called non-singular; it has full rank and an inverse exists. A matrix with det(A) = 0 is singular and has no inverse.

方阵的行列式，记作 det(A) 或 |A|，是一个决定矩阵是否可逆的标量。行列式不为零的矩阵称为非奇异矩阵；它是满秩的，并且存在逆矩阵。而行列式为零的矩阵是奇异矩阵，不存在逆矩阵。

For a 2×2 matrix A = [a b; c d], the determinant is ad - bc. For larger matrices, determinants are computed using expansion by minors, which naturally leads to the concept of cofactors and the adjoint.

对于 2×2 矩阵 A = [a b; c d]，行列式为 ad - bc。对于更大的矩阵，行列式通过子式展开来计算，这自然引出了余子式和伴随矩阵的概念。

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3. Minors and Cofactors | 子式与余子式

The minor M_ij of an element a_ij in a square matrix is the determinant of the submatrix formed by deleting the i-th row and j-th column. The cofactor C_ij is defined as C_ij = (-1)^i+j M_ij. The sign factor (+ or -) creates a checkerboard pattern starting with + in the (1,1) position.

方阵中元素 a_ij 的子式 M_ij 是划去第 i 行和第 j 列后所得子矩阵的行列式。余子式 C_ij 定义为 C_ij = (-1)^i+j M_ij。符号因子（+ 或 -）形成棋盘格图案，从 (1,1) 位置的 + 开始。

Understanding cofactors is crucial because the determinant itself can be expressed as the sum of products of elements from any row or column with their respective cofactors, e.g., det(A) = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ for a 3×3 matrix.

理解余子式至关重要，因为行列式本身可以表示为任意一行或一列的元素与其相应余子式的乘积之和，例如对于 3×3 矩阵，det(A) = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃。

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4. The Matrix of Cofactors | 余子式矩阵

By replacing every element of the original matrix with its cofactor, we obtain the cofactor matrix, denoted cof(A). For an n×n matrix A, the cofactor matrix has the form:

将原矩阵中的每个元素替换为其余子式，我们得到余子式矩阵，记作 cof(A)。对于 n×n 矩阵 A，余子式矩阵的形式为：

C11	C12	…	C1n
C21	C22	…	C2n
⋮	⋮	⋱	⋮
Cn1	Cn2	…	Cnn

This matrix captures all the signed minor data but is not yet directly usable to find the inverse. An additional transpose step is required.

这个矩阵包含了所有带符号的子式信息，但此时还不能直接用于求逆矩阵，还需要一个转置步骤。

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5. The Adjoint (Adjugate) Matrix | 伴随矩阵

The adjoint (or classical adjoint) of A, written adj(A), is the transpose of the cofactor matrix: adj(A) = (cof(A))^T. In other words, the entry in the i-th row and j-th column of adj(A) is C_ji, the cofactor of a_ji.

A 的伴随矩阵（或古典伴随矩阵），记作 adj(A)，是余子式矩阵的转置：adj(A) = (cof(A))^T。也就是说，adj(A) 中第 i 行第 j 列的元素是 C_ji，即 a_ji 的余子式。

This transposition is essential because it ensures the product A · adj(A) yields a diagonal matrix with the determinant repeated on the diagonal. The adjoint matrix can be computed for any square matrix, even singular ones, and its rank reveals information about the original matrix's invertibility.

这一转置至关重要，因为它能使得乘积 A · adj(A) 得到一个对角线上全是行列式值的对角阵。任何方阵（即使是奇异矩阵）都可以计算伴随矩阵，其秩能揭示原矩阵可逆性的信息。

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6. Formula for the Inverse Matrix | 逆矩阵公式

If det(A) ≠ 0, the inverse of A is given by the celebrated formula:

A^-1 = (1 / det(A)) adj(A)

如果 det(A) ≠ 0，A 的逆矩阵由以下著名公式给出：

A^-1 = (1 / det(A)) adj(A)

. For 2×2 matrices, a simplified version exists: if A = [a b; c d] and ad - bc ≠ 0, then A^-1 (1/(ad-bc)) [d -b; -c a], which is precisely the adjoint formula applied to the 2×2 case.

这个公式提供了一种系统化的求逆方法，无需直接解方程组。它将问题转化为求行列式与所有余子式。对于 2×2 矩阵，有一个简化版本：若 A = [a b; c d] 且 ad - bc ≠ 0，则 A^-1 = (1/(ad-bc)) [d -b; -c a]，这正是伴随矩阵公式在 2×2 情形下的应用。

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7. Worked Example: 2×2 Matrix | 范例：2×2 矩阵

s gives cofactor matrix [4 -1; -2 3]. Transposing yields adj(A) = [4 -2; -1 3]. Finally, A^-1 = (1/10) [4 -2; -1 3] = [0.4 -0.2; -0.1 0.3].

设 A = [3 2; 1 4]。首先，det(A) = 3×4 - 2×1 = 12 - 2 = 10 ≠ 0，因此逆矩阵存在。子式矩阵为 [4 1; 2 3]。应用棋盘格符号得到余子式矩阵 [4 -1; -2 3]。转置后得到 adj(A) = [4 -2; -1 3]。最终，A^-1 = (1/10) [4 -2; -1 3] = [0.4 -0.2; -0.1 0.3]。

Multiplying to check: A A^-1 )+2(0.3); 1(0.4)+4(-0.1) 1(-0.2)+4(0.3)] = [1 0; 0 1]. The adjoint method works perfectly.

乘法验算：A A^-1 = [3(0.4)+2(-0.1) 3(-0.2)+2(0.3); 1(0.4)+4(-0.1) 1(-0.2)+4(0.3)] = [1 0; 0 1]。伴随矩阵法完全正确。

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8. Worked Example: 3×3 Matrix | 范例：3×3 矩阵

Consider A =
[1 2 3; 0 1 4; 5 6 0]. First, expand determinant by row 1:
×0 - 4×6) - 2·(0×0 - 4×5) + 3·(0×6 - 1×5) = 1·(0-24) -2·(0-20) +3·(0-5) = -24 +40 -15 = 1.

考虑 A =
[1 2 3; 0 1 4; 5 6 0]。首先按第一行展开行列式：
1×0 - 4×6) - 2·(0×0 - 4×5) + 3·(0×6 - 1×5) = 1·(0-24) -2·(0-20) +3·(0-5) = -24 +40 -15 = 1。

Next, compute all nine cofactors. For example, C₁₁ = + (1×0 - 4×6) = -24, C₁₂ = - (0×0 - 4×5) = +20, C₁₃ = + (0×6 - 1×5) = -5, and so on. The cofactor matrix is:
[-24 20 -5; 18 -15 4; 5 -4 1]. Its transpose – the adjoint – is:
dj(A) = [-24 18 5; 20 -15 -4; -5 4 1]. Since det(A) = 1, the inverse equals the adjoint: A^-1 = adj(A).

接下来计算全部九个余子式。例如 C₁₁ = + (1×0 - 4×6) = -24，C₁₂ = - (0×0 - 4×5) = +20，C₁₃ = + (0×6 - 1×5) = -5，等等。余子式矩阵为：
[-24 20 -5; 18 -15 4; 5 -4 1]。其转置——伴随矩阵——为：
adj(A) = [-24 18 5; 20 -15 -4; -5 4 1]。因为 det(A) = 1，逆矩阵就等于伴随矩阵：A^-1 = adj(A)。

e matrix particularly elegant. Such matrices are called unimodular and appear frequently in geometry and number theory.

这个例子突显了 det(A) = 1 的特殊情形，此时逆矩阵尤为简洁。这类矩阵称为幺模矩阵，在几何和数论中经常出现。

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9. Verification and Properties | 验证与性质

A fundamental identity links a matrix, its adjoint and its determinant: A · adj(A) = adj(A) · A = det(A) · I. This can be derived from the expansion of determinants and the definition of cofactors. If det(A) ≠ 0, dividing through by det(A) immediately gives A A^-1 = I.

一个基本恒等式将矩阵、伴随矩阵及行列式联系起来：A · adj(A) = adj(A) · A = det(A) · I。这可以从行列式展开及余子式的定义推导得到。若 det(A) ≠ 0，等式两边除以 det(A) 即得 A A^-1 = I。

Important properties of the adjoint and inverse include:

• (A^-1)^-1 = A for invertible A.
• (kA)^-1 = (1/k) A^-1 for non-zero scalar k.
• (AB)^-1 = B^-1 A^-1 when both A and B are invertible.
• adj(AB) = adj(B) adj(A) (analogous to the inverse product rule).
• det(adj(A)) = (det(A))^n-1 for an n×n matrix.
• adj(adj(A)) = (det(A))^n-2 A (n ≥ 3).

逆矩阵与伴随矩阵的重要性质包括：

• 若 A 可逆，则 (A^-1)^-1 = A。
• 对于非零标量 k，(kA)^-1 = (1/k) A^-1。
• 当 A 与 B 均可逆时，(AB)^-1 = B^-1 A^-1。
• adj(AB) = adj(B) adj(A)（类似于逆矩阵的乘积法则）。
• 对于 n×n 矩阵，det(adj(A)) = (det(A))^n-1。
• 当 n ≥ 3 时，adj(adj(A)) = (det(A))^n-2 A。

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10. Solving Linear Systems Using Inverses | 利用逆矩阵解线性方程组

One major application of the inverse matrix is solving a system of linear equations written in matrix form AX = B. If A is an invertible n×n matrix, the unique solution is X = A^-1B. Using the adjoint formula, we can express the solution as X = (1 / det(A)) adj(A) B, which directly yields Cramer's rule for each variable.

逆矩阵的一个主要应用是求解矩阵形式的线性方程组 AX = B。若 A 是可逆的 n×n 矩阵，其唯一解为 X = A^-1B。利用伴随矩阵公式，可将解表示为 X = (1 / det(A)) adj(A) B，这直接导出了针对每个变量的克莱姆法则。

For example, the system
oint method ensures a systematic path to the solution, especially useful for small systems.

例如，方程组
2x + y = 5, 3x + 4y = 6 可以写为 A = [2 1; 3 4]，X = [x; y]，B = [5; 6]。因为 det(A) = 5，逆矩阵为 (1/5)[4 -1; -3 2]。于是 X = (1/5)[4 -1; -3 2][5; 6] = (1/5)[14; -3] = [2.8; -0.6]。伴随矩阵法确保了求解的条理性，对小规模方程组尤为实用。

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11. When the Inverse Does Not Exist | 逆矩阵不存在的情形

singular matrix has rank strictly less than n, and its adjoint has rank at most 1 (if n ≥

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