📚 Inverse Matrices | 逆矩阵
es are a central tool in AQA A-level Mathematics and Further Mathematics. They allow us to 'undo' inverse formula, singular matrices, properties, applications, and common exam techniques.
逆矩阵是 AQA A-level 数学及进阶数学中的核心工具。它们让我们能够“撤销”一个矩阵变换，并求解写成矩阵形式的线性方程组。本文涵盖 2×2 逆矩阵公式、奇异矩阵、性质、应用以及常见考试技巧。
1. What Is an Inverse Matrix? | 什么是逆矩阵？
of the same order. For a 2×2 matrix, I = [1 0; 0 1]. The inverse is unique when it exists.
对于一个方阵 A，其逆矩阵记作 A⁻¹。它满足 AA⁻¹ = A⁻¹A = I，其中 I 是同阶单位矩阵。对于 2×2 矩阵，I = [1 0; 0 1]。逆矩阵如果存在，则是唯一的。
2. The 2×2 Inverse Formula | 2×2 逆矩阵公式
= [a b; c d], the determinant is det A = ad − bc. When det A ≠ 0, the inverse is given by:
如果 A = [a b; c d]，其行列式为 det A = ad − bc。当 det A ≠ 0 时，逆矩阵公式为：
A⁻¹ = 1/(ad − bc) × [d −b; −c a]
In words: swap the entries a and d, change the signs of b and c, then divide by the determinant.
用语言描述：交换 a 和 d 的位置，将 b 和 c 变号，然后除以行列式。
3. Determinant and Singular Matrices | 行列式与奇异矩阵
, the matrix is singular and has no inverse. This usually means the two rows are proportional or the transformation collapses the plane onto a line or point.
行列式 det A = ad − bc 决定逆矩阵是否存在。如果 det A = 0，该矩阵是奇异矩阵，没有逆矩阵。这通常意味着两行成比例，或者该变换将平面压缩到一条直线或一个点。
- det A ≠ 0: non-singular, A⁻¹ exists. | det A ≠ 0：非奇异，A⁻¹ 存在。
- det A = 0: singular, A⁻¹ does not exist. | det A = 0：奇异，A⁻¹ 不存在。
4. Worked Example: Finding a 2×2 Inverse | 例题：求 2×2 逆矩阵
Find A⁻¹ for A = [4 7; 2 6].
求 A = [4 7; 2 6] 的逆矩阵 A⁻¹。
First calculate det A = 4×6 − 7×2 = 24 − 14 = 10. Since det A ≠ 0, A is non-singular.
首先计算 det A = 4×6 − 7×2 = 24 − 14 = 10。因为 det A ≠ 0，所以 A 非奇异。
A⁻¹ = 1/10 × [6 −7; −2 4]
This can be written as [3/5 −7/10; −1/5 2/5].
这可以写成 [3/5 −7/10; −1/5 2/5]。
5. Verifying an Inverse | 验证逆矩阵
To check an inverse, multiply A by A⁻¹ in both orders. For the example above:
要检验逆矩阵，需要用两种顺序计算 A 与 A⁻¹ 的乘积。以上例为例：
A × A⁻¹ = [4 7; 2 6] × [3/5 −7/10; −1/5 2/5] = [1 0; 0 1]
= I, although for square matrices one side is enough to guarantee the other at this level.
乘积是单位矩阵，因此逆矩阵正确。你还应该验证 A⁻¹A = I，不过在本阶段的方阵中，验证一侧通常就足以保证另一侧成立。
6. Properties of Inverse Matrices | 逆矩阵的性质
Useful algebraic properties include:
常用的代数性质包括：
- (A⁻¹)⁻¹ = A — the inverse of an inverse is the original matrix. | (A⁻¹)⁻¹ = A —— 逆矩阵的逆是原矩阵。
- (kA)⁻¹ = 1/k × A⁻¹ for a non-zero scalar k. | 对于非零标量 k，(kA)⁻¹ = 1/k × A⁻¹。
- (Aᵀ)⁻¹ = (A⁻¹)ᵀ — the transpose and inverse operations commute. | (Aᵀ)⁻¹ = (A⁻¹)ᵀ —— 转置与求逆可以交换顺序。
- det(A⁻¹) = 1/det(A). | det(A⁻¹) = 1/det(A)。
These properties are frequently tested in AQA questions that require simplifying matrix expressions.
这些性质在 AQA 的矩阵表达式化简题中经常出现。
7. Inverse of a Product | 乘积的逆矩阵
For two invertible matrices A and B of the same order, the inverse of the product reverses the order:
对于两个同阶可逆矩阵 A 和 B，乘积的逆矩阵需要交换顺序：
(AB)⁻¹ = B⁻¹A
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