Inverse Matrices | 逆矩阵

📚 Inverse Matrices | 逆矩阵

In matrix algebra, the inverse of a matrix plays a role analogous to the reciprocal of a number. For a square matrix A, its inverse A⁻¹ satisfies AA⁻¹ = A⁻¹A = I, where I is the identity matrix. This concept is fundamental for solving linear systems and understanding linear transformations.

在矩阵代数中，逆矩阵的作用类似于数字的倒数。对于方阵 A，其逆矩阵 A⁻¹ 满足 AA⁻¹ = A⁻¹A = I，其中 I 为单位矩阵。这一概念是解线性方程组和理解线性变换的基础。

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

Let A be a square matrix of order n. If there exists a square matrix B of the same order such that AB = BA = Iₙ, then B is called the inverse of A and is denoted by A⁻¹. Not every matrix has an inverse; a matrix that possesses an inverse is said to be invertible or non-singular.

设 A 为 n 阶方阵。若存在同阶方阵 B，使得 AB = BA = Iₙ，则称 B 为 A 的逆矩阵，记为 A⁻¹。并非每个矩阵都有逆矩阵；有逆矩阵的矩阵称为可逆矩阵或非奇异矩阵。

The identity matrix Iₙ has 1s on the main diagonal and 0s elsewhere. For example, for order 2:

单位矩阵 Iₙ 的主对角线元素为 1，其余元素为 0。例如，二阶单位矩阵为：

I₂ = [ 1 0 ; 0 1 ]

If A⁻¹ exists, it is unique. That is, there cannot be two different matrices that both act as the inverse of A.

若 A⁻¹ 存在，则它唯一。也就是说，不可能存在两个不同的矩阵同时作为 A 的逆矩阵。

2. Existence and Uniqueness | 存在性与唯一性

A square matrix A has an inverse if and only if its determinant is non-zero, i.e. det(A) ≠ 0. If det(A) = 0, the matrix is called singular and has no inverse. The determinant acts as a measure of whether the rows or columns are linearly independent.

方阵 A 存在逆矩阵当且仅当其行列式不为零，即 det(A) ≠ 0。若 det(A) = 0，则称该矩阵为奇异矩阵，它没有逆矩阵。行列式可以衡量矩阵的行或列是否线性无关。

For a 2×2 matrix A = [ a b ; c d ], the determinant is computed as:

对于二阶矩阵 A = [ a b ; c d ]，行列式的计算公式为：

det(A) = ad − bc

If ad − bc = 0, no inverse exists. This examination of the determinant is always the first step when finding an inverse.

若 ad − bc = 0，则逆矩阵不存在。在求逆矩阵时，检查行列式永远是第一步。

3. The Inverse of a 2×2 Matrix | 2×2 矩阵的逆

Given a 2×2 matrix A = [ a b ; c d ] with det(A) = ad − bc ≠ 0, the inverse is given by:

已知二阶矩阵 A = [ a b ; c d ]，且 det(A) = ad − bc ≠ 0，其逆矩阵为：

A⁻¹ = (1 / (ad − bc)) × [ d −b ; −c a ]

Notice that the diagonal entries a and d are swapped, and the off-diagonal entries b and c change sign. Then every entry is divided by the determinant.

注意，主对角线上的元素 a 和 d 交换位置，副对角线上的元素 b 和 c 改变符号，然后每一项都除以行列式。

Example: For A = [ 2 3 ; 1 4 ], det(A) = 2×4 − 3×1 = 5. Hence:

例如：对于 A = [ 2 3 ; 1 4 ]，det(A) = 2×4 − 3×1 = 5。因此：

A⁻¹ = (1/5) × [ 4 −3 ; −1 2 ] = [ 0.8 −0.6 ; −0.2 0.4 ]

Always verify by multiplying A and A⁻¹; the result should be the identity matrix.

务必通过相乘 A 和 A⁻¹ 进行验证，结果应为单位矩阵。

4. Adjugate and Determinant for 3×3 Matrices | 3×3 矩阵的伴随矩阵与行列式

For a 3×3 matrix, the inverse formula generalises as:

对于三阶矩阵，逆矩阵公式推广为：

A⁻¹ = (1 / det(A)) × adj(A)

where adj(A) is the adjugate (or adjoint) matrix, formed by the cofactor matrix followed by a transpose. The determinant of a 3×3 matrix can be computed using the rule of Sarrus or by expansion along a row or column.

其中 adj(A) 为伴随矩阵，由代数余子式矩阵取转置得到。三阶行列式可以使用萨鲁斯法则或按某一行（列）展开来计算。

To find the cofactor Cᵢⱼ of entry aᵢⱼ, delete row i and column j, take the determinant of the remaining 2×2 submatrix, and multiply by (−1)ⁱ⁺ʲ.

若要计算元素 aᵢⱼ 的代数余子式 Cᵢⱼ，先删去第 i 行和第 j 列，求剩余 2×2 子矩阵的行列式，再乘以 (−1)ⁱ⁺ʲ。

For example, given A = [ a b c ; d e f ; g h i ], det(A) = a(ei − fh) − b(di − fg) + c(dh − eg).

例如，设 A = [ a b c ; d e f ; g h i ]，则 det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)。

Then the cofactor matrix is formed by all nine cofactors, and its transpose gives the adjugate. Finally, multiply by 1/det(A).

然后由九个代数余子式构成余子式矩阵，取转置得到伴随矩阵，最后乘以 1/det(A) 即得逆矩阵。

5. Using Row Reduction to Find the Inverse | 用行化简求逆

An alternative method for finding the inverse of a matrix is Gauss-Jordan elimination. Place the matrix A and the identity matrix I side by side: [ A | I ]. Then apply elementary row operations to transform the left side into I. Once this is done, the right side becomes A⁻¹.

求逆矩阵的另一种方法是高斯-约当消元法。将矩阵 A 和单位矩阵 I 并排放置为 [ A | I ]，然后对行进行初等行变换，使左侧变为 I。一旦完成，右侧即为 A⁻¹。

This method is efficient for 3×3 and larger matrices. It also works when the determinant is zero, but in that case the row reduction will produce a row of zeros on the left, indicating that no inverse exists.

这种方法对于三阶及更大的矩阵十分有效。当行列式为零时，行化简会在左侧产生全零行，从而说明逆矩阵不存在。

Example: For A = [ 1 2 ; 3 4 ], write [ 1 2 | 1 0 ; 3 4 | 0 1 ]. Row operations yield:

例如：对 A = [ 1 2 ; 3 4 ]，写出 [ 1 2 | 1 0 ; 3 4 | 0 1 ]。经过行变换得到：

[ 1 0 | −2 1 ; 0 1 | 1.5 −0.5 ]

Thus A⁻¹ = [ −2 1 ; 1.5 −0.5 ], which matches the formula result because det(A) = 1×4 − 2×3 = −2.

因此 A⁻¹ = [ −2 1 ; 1.5 −0.5 ]，这与公式结果一致，因为 det(A) = 1×4 − 2×3 = −2。

6. Key Properties of Inverse Matrices | 逆矩阵的重要性质

The inverse operation has several useful properties that simplify calculations in matrix algebra.

逆运算具有若干重要性质，可以简化矩阵代数中的计算。

(A⁻¹)⁻¹ = A — the inverse of the inverse gives back the original matrix.

(A⁻¹)⁻¹ = A —— 逆矩阵的逆矩阵等于原矩阵。
(AB)⁻¹ = B⁻¹A⁻¹ — the inverse of a product reverses the order.

(AB)⁻¹ = B⁻¹A⁻¹ —— 乘积的逆矩阵要交换顺序。
(Aᵀ)⁻¹ = (A⁻¹)ᵀ — the transpose and the inverse commute.

(Aᵀ)⁻¹ = (A⁻¹)ᵀ —— 转置与求逆可以交换次序。
det(A⁻¹) = 1/det(A) — the determinant of the inverse is the reciprocal of the determinant.

det(A⁻¹) = 1/det(A) —— 逆矩阵的行列式是原矩阵行列式的倒数。

These properties are frequently tested in AQA examinations, especially the reverse order in (AB)⁻¹.

这些性质在 AQA 考试中经常出现，尤其是 (AB)⁻¹ 中矩阵顺序颠倒这一点。

7. Solving Linear Systems with the Inverse | 用逆矩阵解线性方程组

Consider a system of linear equations written in matrix form as AX = B, where A is the coefficient matrix, X is the column vector of unknowns, and B is the constant vector. If A is invertible, multiply both sides on the left by A⁻¹:

考虑以矩阵形式 AX = B 表示的线性方程组，其中 A 是系数矩阵，X 是未知数列向量，B 是常数向量。若 A 可逆，则在等式两边左乘 A⁻¹：

A⁻¹AX = A⁻¹B ⇒ X = A⁻¹B

Because A⁻¹A = I, the unknown vector X is directly obtained. This method is particularly clear for 2×2 systems.

由于 A⁻¹A = I，未知向量 X 可直接得出。这种方法在二阶方程组中尤其清晰。

Example: Solve 2x + 3y = 7 and x + 4y = 6. Here A = [ 2 3 ; 1 4 ], X = [ x ; y ], B = [ 7 ; 6 ]. From earlier, A⁻¹ = (1/5)[ 4 −3 ; −1 2 ]. Therefore:

例如：解方程组 2x + 3y = 7 和 x + 4y = 6。这里 A = [ 2 3 ; 1 4 ]，X = [ x ; y ]，B = [ 7 ; 6 ]。由前例可知 A⁻¹ = (1/5)[ 4 −3 ; −1 2 ]。因此：

X = (1/5)[ 4 −3 ; −1 2 ][ 7 ; 6 ] = (1/5)[ 28−18 ; −7+12 ] = (1/5)[ 10 ; 5 ] = [ 2 ; 1 ]

So x = 2 and y = 1.

所以 x = 2，y = 1。

8. Inverse of a Product | 乘积的逆

The formula (AB)⁻¹ = B⁻¹A⁻¹ deserves special attention. To verify it, compute:

公式 (AB)⁻¹ = B⁻¹A⁻¹ 值得特别注意。验证方法如下：

(AB)(B⁻¹A⁻¹) = A(BB⁻¹)A⁻¹ = AIA⁻¹ = AA⁻¹ = I

Similarly, multiplying B⁻¹A⁻¹ by AB on the right also gives I. Hence the order must be reversed. A common mistake is to write A⁻¹B⁻¹, which is generally wrong.

类似地，用 B⁻¹A⁻¹ 右乘 AB 也得到 I。因此顺序必须颠倒。常见的错误是写成 A⁻¹B⁻¹，这通常是错误的。

This property extends to more factors: (ABC)⁻¹ = C⁻¹B⁻¹A⁻¹.

这一性质可以推广到更多因子的情形：(ABC)⁻¹ = C⁻¹B⁻¹A⁻¹。

9. Inverse of a Transpose | 转置的逆

For any invertible matrix A, the inverse of its transpose equals the transpose of its inverse:

对于任意可逆矩阵 A，其转置的逆等于其逆的转置：

(Aᵀ)⁻¹ = (A⁻¹)ᵀ

This identity is useful when working with symmetric matrices or when solving systems with transpose coefficients. If A is symmetric, then Aᵀ = A, and therefore A⁻¹ is also symmetric.

该恒等式在处理对称矩阵或具有转置系数的方程组时非常有用。若 A 是对称矩阵，则 Aᵀ = A，因此 A⁻¹ 也是对称的。

10. Common Pitfalls and Exam Tips | 常见错误与考试技巧

Students often make avoidable errors when computing inverses. Below are the most frequent pitfalls and how to avoid them.

学生在计算逆矩阵时经常犯可避免的错误。以下是最常见的陷阱以及避免方法。

Forgetting to check det(A) ≠ 0 before attempting to find an inverse. Always compute the determinant first.

在尝试求逆之前忘记检查 det(A) ≠ 0。务必先计算行列式。
Misplacing signs in the 2×2 inverse: the diagonal entries are swapped, and the off-diagonal entries change sign. Review the formula each time.

在二阶逆矩阵中放错符号：主对角线元素交换，副对角线元素变号。每次都要核对公式。
Incorrectly ordering the product in (AB)⁻¹. Use B⁻¹A⁻¹, not A⁻¹B⁻¹.

在 (AB)⁻¹ 中错误地排列顺序。应使用 B⁻¹A⁻¹，而不是 A⁻¹B⁻¹。
When solving AX = B, multiply on the left by A⁻¹ to get X = A⁻¹B. Multiplying on the right would give XAA⁻¹, which is not valid because X and B are column vectors.

解 AX = B 时，应在左方乘以 A⁻¹ 得到 X = A⁻¹B。如果在右方乘，会得到 XAA⁻¹，这不对，因为 X 和 B 是列向量。
Arithmetic errors when calculating cofactors for 3×3 matrices. Work step by step and check signs.

计算三阶矩阵的代数余子式时出现算术错误。要逐步计算并检查符号。

In examinations, always show your determinant calculation clearly, as method marks are awarded even for small mistakes in the final inverse. Practice with past paper questions to build speed and accuracy.

考试中务必清晰展示行列式的计算过程，因为即使最终逆矩阵有少量错误，方法分仍然可以获得。通过练习真题来提高速度和准确性。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Inverse Matrices | 逆矩阵