Inverse Matrices | 逆矩阵

📚 Inverse Matrices | 逆矩阵

Inverse matrices are a cornerstone of linear algebra in A-Level mathematics. They allow us to ‘undo’ a matrix transformation and provide a powerful method for solving systems of simultaneous equations. This article covers everything you need for the AQA specification: definitions, formulas, worked examples, and exam-style tips.

逆矩阵是A-Level数学中线性代数的基石。它使我们能够”撤销”一个矩阵变换，并为求解线性方程组提供了一种强大的方法。本文将涵盖AQA考纲所需的全部内容：定义、公式、实例演练和考试技巧。

1. What Is an Inverse Matrix? | 什么是逆矩阵？

For a square matrix A, its inverse is denoted A⁻¹ and is the unique matrix that satisfies the equation A × A⁻¹ = A⁻¹ × A = I, where I is the identity matrix of the same order.

对于一个方阵A，其逆矩阵记为A⁻¹，它是满足等式A × A⁻¹ = A⁻¹ × A = I的唯一矩阵，其中I是同阶单位矩阵。

A × A⁻¹ = A⁻¹ × A = I

Just as multiplying a number by its reciprocal (e.g., 5 × 1⁄5 = 1) gives 1, multiplying a matrix by its inverse gives the identity matrix. Not every matrix has an inverse; a matrix must be square and have a non-zero determinant.

就像数字乘以它的倒数（例如5 × 1⁄5 = 1）得到1一样，矩阵乘以它的逆矩阵得到单位矩阵。并非所有矩阵都有逆矩阵；矩阵必须是方阵且行列式不为零。

2. The Identity Matrix | 单位矩阵

The identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere. It acts as the multiplicative identity: multiplying any matrix by the identity leaves it unchanged.

单位矩阵是主对角线上的元素为1、其余位置为0的方阵。它起着乘法单位元的作用：任何矩阵乘以单位矩阵都保持不变。

I₂ = [1 0; 0 1], I₃ = [1 0 0; 0 1 0; 0 0 1]

For example, if A is a 2×2 matrix:

例如，若A是一个2×2矩阵：

A × I₂ = I₂ × A = A

The identity matrix is essential in verifying that a candidate matrix is indeed the inverse of another matrix. If your product equals the identity, you have found the correct inverse.

单位矩阵在验证一个候选矩阵是否确实是另一个矩阵的逆矩阵时至关重要。如果乘积等于单位矩阵，说明你找到了正确的逆矩阵。

3. When Does an Inverse Exist? | 逆矩阵存在的条件

Two conditions must be met for a matrix to have an inverse:

矩阵存在逆矩阵必须满足两个条件：

The matrix must be square (same number of rows and columns). 矩阵必须是方阵（行数和列数相等）。
The determinant must be non-zero (det(A) ≠ 0). 行列式必须不为零（det(A) ≠ 0）。

A matrix with a zero determinant is called singular or non-invertible. A matrix with a non-zero determinant is called non-singular or invertible.

行列式为零的矩阵称为奇异矩阵或不可逆矩阵。行列式不为零的矩阵称为非奇异矩阵或可逆矩阵。

Geometrically, a singular matrix maps a 2D shape to a line or a point, collapsing the area to zero. An invertible matrix, by contrast, preserves a non-zero area (its determinant equals the scale factor of that area).

从几何上看，奇异矩阵将二维图形映射为一条直线或一个点，把面积压缩为零。相反，可逆矩阵保持非零面积（其行列式等于该面积的缩放因子）。

4. Inverse of a 2×2 Matrix | 2×2矩阵的逆矩阵

For a 2×2 matrix

对于一个2×2矩阵

A = [a b; c d]

its inverse is given by the formula:

其逆矩阵由以下公式给出：

A⁻¹ = 1⁄(ad − bc) × [d −b; −c a]

Here, ad − bc is the determinant of A, written as det(A). The steps are:

这里，ad − bc是矩阵A的行列式，记作det(A)。步骤如下：

Swap the entries on the main diagonal: a and d are swapped. 交换主对角线上的元素：a和d互换。
Change the signs of the off-diagonal entries: b and c become −b and −c. 改变副对角线元素的符号：b和c变为−b和−c。
Multiply the resulting matrix by 1⁄det(A). 将结果矩阵乘以1⁄det(A)。

This formula is given in the AQA formula booklet, but you must know how to apply it quickly and accurately under exam conditions.

这个公式在AQA公式册中给出，但你必须能在考试条件下快速准确地应用它。

5. Worked Example | 实例演练

Find the inverse of the matrix A = [2 3; 1 4].

求矩阵A = [2 3; 1 4]的逆矩阵。

Step 1: Calculate the determinant.

第一步：计算行列式。

det(A) = 2 × 4 − 3 × 1 = 8 − 3 = 5

Step 2: Apply the 2×2 inverse formula.

第二步：应用2×2逆矩阵公式。

A⁻¹ = 1⁄5 × [4 −3; −1 2]

A⁻¹ = [4⁄5 −3⁄5; −1⁄5 2⁄5]

Step 3: Verify by multiplying A and A⁻¹.

第三步：通过乘法验证A和A⁻¹。

[2 3; 1 4] × [4⁄5 −3⁄5; −1⁄5 2⁄5] = [1 0; 0 1]

The result confirms our answer is correct. Always perform a quick check — it only takes seconds and can prevent careless errors.

结果证实我们的答案是正确的。务必快速检验——只需几秒钟，却能避免粗心错误。

6. Solving Simultaneous Equations | 解联立方程组

Inverse matrices provide an elegant method for solving systems of linear equations. Consider the system:

逆矩阵为求解线性方程组提供了一种优雅的方法。考虑以下方程组：

2x + 3y = 7
x + 4y = 6

This can be written in matrix form as AX = B:

这可以写成矩阵形式AX = B：

[2 3; 1 4] [x; y] = [7; 6]

To solve for X, multiply both sides by A⁻¹ on the left:

为了求解X，在等式两边左乘A⁻¹：

X = A⁻¹B

Using the inverse we found earlier:

利用我们之前求得的逆矩阵：

X = [4⁄5 −3⁄5; −1⁄5 2⁄5] [7; 6]

X = [28⁄5 − 18⁄5; −7⁄5 + 12⁄5] = [10⁄5; 5⁄5] = [2; 1]

Therefore, the solution is x = 2, y = 1. This method is particularly efficient for systems where the same coefficient matrix is used repeatedly.

因此，解为x = 2, y = 1。当多个方程组共用同一个系数矩阵时，这种方法尤为高效。

7. Properties of Inverse Matrices | 逆矩阵的性质

The following properties are essential for solving problems and simplifying expressions:

以下性质对于解题和化简表达式至关重要：

The inverse is unique. If a matrix is invertible, there is exactly one inverse. 逆矩阵是唯一的。若矩阵可逆，则逆矩阵唯一存在。
The inverse of the inverse is the original matrix: (A⁻¹)⁻¹ = A. 逆矩阵的逆是原矩阵：(A⁻¹)⁻¹ = A。
The determinant of the inverse is the reciprocal: det(A⁻¹) = 1⁄det(A). 逆矩阵的行列式是原行列式的倒数：det(A⁻¹) = 1⁄det(A)。
The inverse commutes with the identity: A × A⁻¹ = A⁻¹ × A = I. 逆矩阵与单位矩阵可交换：A × A⁻¹ = A⁻¹ × A = I。

These properties often appear in exam questions that ask you to simplify matrix expressions or prove relationships between matrices.

这些性质经常出现在考试题目中，要求你化简矩阵表达式或证明矩阵之间的关系。

8. Inverse of a Product | 乘积的逆矩阵

If A and B are both invertible matrices of the same order, then the inverse of their product is the product of their inverses in reverse order:

若A和B都是同阶可逆矩阵，则它们乘积的逆矩阵等于它们各自的逆矩阵按相反顺序相乘：

(AB)⁻¹ = B⁻¹A⁻¹

The order matters because matrix multiplication is not commutative. Let us verify this identity:

顺序很重要，因为矩阵乘法不满足交换律。让我们验证这个恒等式：

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

Since multiplying AB by B⁻¹A⁻¹ yields the identity matrix, the inverse of AB is indeed B⁻¹A⁻¹.

由于AB乘以B⁻¹A⁻¹得到单位矩阵，所以AB的逆矩阵确实是B⁻¹A⁻¹。

A common mistake is to write (AB)⁻¹ = A⁻¹B⁻¹. This is incorrect — always reverse the order.

一个常见错误是写成(AB)⁻¹ = A⁻¹B⁻¹。这是错误的——始终要反序相乘。

9. Singular Matrices and Special Cases | 奇异矩阵与特殊情况

A singular matrix has a determinant of zero and therefore has no inverse. In the 2×2 case, this means ad − bc = 0.

奇异矩阵的行列式为零，因此没有逆矩阵。在2×2情形下，这意味着ad − bc = 0。

For example, the matrix [2 4; 1 2] is singular because:

例如，矩阵[2 4; 1 2]是奇异的，因为：

det(A) = 2 × 2 − 4 × 1 = 4 − 4 = 0

When using the formula, you would be attempting to divide by zero, which is undefined. Geometrically, a singular matrix collapses the plane into a line, so the transformation cannot be reversed.

当使用公式时，你相当于试图除以零，这是无定义的。从几何上看，奇异矩阵把平面压缩成一条线，因此该变换无法逆转。

In simultaneous equations, a singular coefficient matrix indicates that the system either has no solution or infinitely many solutions (the equations are dependent or inconsistent).

在线性方程组中，奇异系数矩阵表示方程组要么无解，要么有无穷多解（方程相关或矛盾）。

10. Transformations and Inverse Matrices | 变换与逆矩阵

A 2×2 matrix can represent a linear transformation such as reflection, rotation, or enlargement. The inverse matrix represents the transformation that reverses the original one.

2×2矩阵可以表示线性变换，例如反射、旋转或缩放。逆矩阵则表示逆转原始变换的变换。

For instance, a rotation of 90° anticlockwise is given by:

例如，逆时针旋转90°的变换矩阵为：

R = [0 −1; 1 0]

Its inverse should represent a rotation of 90° clockwise:

其逆矩阵应当表示顺时针旋转90°：

R⁻¹ = 1⁄(0×0 − (−1)×1) × [0 1; −1 0]

R⁻¹ = 1⁄1 × [0 1; −1 0] = [0 1; −1 0]

Indeed, R × R⁻¹ = [0 −1; 1 0] [0 1; −1 0] = [1 0; 0 1], confirming the identities of the transformation pairs.

确实，R × R⁻¹ = [0 −1; 1 0] [0 1; −1 0] = [1 0; 0 1]，证实了这对变换互逆。

When asked to find the matrix of the reverse transformation, simply compute the inverse of the forward transformation matrix.

当题目要求求逆向变换的矩阵时，只需计算正向变换矩阵的逆矩阵即可。

11. Common Mistakes and Exam Tips | 常见错误与考试技巧

Here are the most frequent pitfalls that cost students marks:

以下是学生最常丢分的陷阱：

Forgetting to divide by the determinant. The formula requires multiplying by 1⁄det(A). Omitting this step is the single most common error. 忘记除以行列式。公式要求乘以1⁄det(A)，省略这一步是最常见的错误。
Getting the sign changes wrong. Remember only the main diagonal is swapped; the off-diagonal entries change sign. 符号改变错误。记住只有主对角线交换，副对角线改变符号。
Reversing the order in (AB)⁻¹. Always write (AB)⁻¹ = B⁻¹A⁻¹. 在(AB)⁻¹中弄反顺序。始终写成(AB)⁻¹ = B⁻¹A⁻¹。
Skipping the verification step. A quick multiplication check takes seconds and prevents arithmetic errors. 跳过验证步骤。快速乘法检验只需几秒钟，可以防止运算错误。
Confusing row/column ordering in simultaneous equations. Make sure the constant matrix B is placed correctly: X = A⁻¹B, not BA⁻¹. 在方程组中混淆行列顺序。确保常数矩阵B的位置正确：X = A⁻¹B，而不是BA⁻¹。

In the exam, if you obtain a determinant of zero, stop and reconsider: the inverse does not exist. Do not attempt to divide by zero.

在考试中，如果你算出行列式为零，请停下来重新思考：逆矩阵不存在。不要试图除以零。

12. Practice Questions | 练习题

Try these questions yourself before checking the answers below.

请先自己尝试以下题目，再核对下面的答案。

Question 1: Find the inverse of A = [5 2; 3 1].

题目1：求A = [5 2; 3 1]的逆矩阵。

Question 2: Use an inverse matrix to solve the simultaneous equations:

题目2：用逆矩阵求解以下联立方程组：

5x + 2y = 9
3x + y = 4

Question 3: Determine whether B = [6 3; 4 2] is invertible. Justify your answer.

题目3：判断B = [6 3; 4 2]是否可逆，并说明理由。

Answers:

答案：

Answer 1: det(A) = 5×1 − 2×3 = 5 − 6 = −1, so A⁻¹ = −1 × [1 −2; −3 5] = [−1 2; 3 −5].

答案1：det(A) = 5×1 − 2×3 = 5 − 6 = −1，所以A⁻¹ = −1 × [1 −2; −3 5] = [−1 2; 3 −5]。

Answer 2: Using X = A⁻¹B, we have X = [−1 2; 3 −5] [9; 4] = [−9 + 8; 27 − 20] = [−1; 7]. Hence x = −1, y = 7.

答案2：利用X = A⁻¹B，可得X = [−1 2; 3 −5] [9; 4] = [−9 + 8; 27 − 20] = [−1; 7]。因此x = −1, y = 7。

Answer 3: det(B) = 6×2 − 3×4 = 12 − 12 = 0, so B is singular and therefore not invertible.

答案3：det(B) = 6×2 − 3×4 = 12 − 12 = 0，所以B是奇异矩阵，因此不可逆。

13. Summary | 总结

Inverse matrices are a powerful tool in A-Level mathematics. You must be able to:

逆矩阵是A-Level数学中的一个强大工具。你必须能够：

Compute the determinant of a 2×2 matrix. 计算2×2矩阵的行列式。
Apply the inverse formula correctly, including the swap and sign-change steps. 正确应用逆矩阵公式，包括交换和变号步骤。
Recognise when a matrix is singular and its inverse does not exist. 识别矩阵何时是奇异的，以及何时逆矩阵不存在。
Use inverse matrices to solve simultaneous equations via X = A⁻¹B. 使用逆矩阵通过X = A⁻¹B求解联立方程组。
Apply algebraic properties such as (AB)⁻¹ = B⁻¹A⁻¹. 应用代数性质，如(AB)⁻¹ = B⁻¹A⁻¹。

Master these skills, and inverse matrix questions in the exam will become straightforward marks. Practice regularly, verify your work, and you will build confidence in this topic.

掌握这些技能后，考试中的逆矩阵题目将成为送分题。定期练习、验证结果，你就能在这一主题上建立信心。

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Inverse Matrices | 逆矩阵