2x2 Determinants: AQA A-Level Further Maths | 二阶行列式：AQA A-Level 进阶数学

📚 2x2 Determinants: AQA A-Level Further Maths | 二阶行列式：AQA A-Level 进阶数学

u whether a matrix is invertible, how a linear transformation scales area, and whether two vectors in the plane are linearly dependent. This article covers the definition, the formula det M = ad − bc, worked examples, the geometric meaning as an area scale factor, the key algebraic properties, singular matrices, area problems, typical AQA question types and the exam technique you need to avoid losing marks.

在 AQA A-Level 进阶数学中，二阶矩阵的行列式是矩阵与变换板块最先接触的工具之一。它可以判断矩阵是否可逆，描述线性变换对面积的缩放，也能判断平面上两个向量是否线性相关。本文涵盖二阶行列式的定义、公式 det M = ad − bc、例题详解、作为面积缩放因子的几何意义、关键代数性质、奇异矩阵、面积问题、AQA 常见题型，以及避免失分的应试技巧。

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1. What Is a 2x2 Determinant? | 什么是二阶行列式？

square matrix. For a 2x2 matrix M = [ a b ; c d ], the determinant is written as det M or | M | the matrix itself; it is a scalar that summarises important information about the matrix.

行列式是由方阵计算出的一个数。对于二阶矩阵 M = [ a b ; c d ]，行列式写作 det M 或 | M |。它与矩阵本身不同，是一个标量，概括了矩阵的重要信息。

Two consequences follow immediately: if det M ≠ 0 the matrix is invertible, and if det M = 0 the matrix is singular and collapses the plane onto a line or a single point.

行列式可以告诉我们矩阵是否有逆矩阵；在变换几何中，它告诉我们矩阵放大或缩小面积的倍数。由此立刻得到两个结论：若 det M ≠ 0，矩阵可逆；若 det M = 0，矩阵为奇异矩阵，会把平面压到一条直线或一个点上。

because the definition requires the same number of rows and columns. This is why the topic begins with square matrices such as 2x2 and later extends to 3x3.

每个方阵都有行列式，而非方阵没有。2 × 3 矩阵不存在行列式，因为定义要求行数与列数相同。这就是本主题从二阶方阵入手，之后再扩展到三阶方阵的原因。

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2. Notation and the Basic Formula | 记号与基本公式

For a general 2x2 matrix M = [ a b ; c d ], the determinant is calculated as follows.

对于一般二阶矩阵 M = [ a b ; c d ]，行列式按如下方式计算。

det M = ad − bc

The notation | a b ; c d | means exactly the same as det M. The vertical bars do not mean absolute value; det M may be positive, negative or zero.

记号 | a b ; c d | 与 det M 含义完全相同。竖线并不表示绝对值；det M 可以是正数、负数或零。

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You must multiply the leading diagonal entries a and d, then subtract the product of the other diagonal entries b and c. A useful memory aid is "main diagonal minus anti-diagonal".

你必须先乘主对角线上的元素 a 和 d，再减去另一条对角线上元素 b 和 c 的乘积。一个实用的口诀是"主对角线减副对角线"。

d. Reversing the subtraction changes the sign of the answer, which is one of the most common careless errors in the exam.

顺序很重要。det M = ad − bc，而不是 bc − ad。把减法的顺序写反会使答案变号，这是考试中最常见的粗心错误之一。

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3. Worked Example: Evaluating a 2x2 Determinant | 例题：计算二阶行列式

et M for M = [3 5; 2 7]. Using det M = ad − bc gives det M = 3 × 7 − 5 × 2 = 21 − 10 = 11.

求 M = [3 5; 2 7] 的 det M。使用 det M = ad − bc，得 det M = 3 × 7 − 5 × 2 = 21 − 10 = 11。

As a second example, take N = [1 4; 3 2]. Then det N = 1 × 2 − 4 × 3 = 2 − 12 = −10.

再举一例，对于 N = [1 4; 3 2]，则 det N = 1 × 2 − 4 × 3 = 2 − 12 = −10。

A negative determinant is perfectly valid and usually indicates a reflection in the transformation. It does not mean that the matrix has no inverse, and it does not mean the calculation is wrong.

负的行列式完全合法，通常表示该变换中含有反射。它并不意味着矩阵没有逆矩阵，也不代表计算错误。

tive entries appear, because (−5) × (−4) = +20, so subtracting it gives a negative result.

P = [2 −5; −4 3] 的行列式是 2 × 3 − (−5) × (−4) = 6 − 20 = −14。出现负数元素时必须小心符号，因为 (−5) × (−4) = +20，减去它后结果仍为负。

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4. Determinants and Invertibility | 行列式与可逆性

A 2x2 matrix M has an inverse M⁻¹ if and only if det M ≠ 0. When the determinant is non-zero, the inverse is built from the determinant and a rearrangement of the entries.

二阶矩阵 M 有逆矩阵 M⁻¹ 的充要条件是 det M ≠ 0。当行列式不为零时，

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