📚 Determinants: 2×2 and 3×3 Matrices | 行列式：2×2 与 3×3 矩阵
In AQA A-Level Mathematics and Further Mathematics, determinants are a central tool when working with square matrices. They tell you whether a matrix is invertible, help you calculate inverse matrices, and explain what a linear transformation does to area or volume.
在 AQA A-Level 数学与进阶数学中，行列式是处理方阵的核心工具。它们可以判断矩阵是否可逆、帮助计算逆矩阵，并从几何上解释线性变换对面积或体积的影响。
1. What is a determinant? | 什么是行列式
x has a 3×3 determinant, and so on. The determinant of A is written as det(A) or sometimes |A|.
行列式是从方阵计算出的一个标量值。它只对方阵有定义：2×2 矩阵有 2×2 行列式，3×3 矩阵有 3×3 行列式，依此类推。矩阵 A 的行列式记作 det(A)，有时也写作 |A|。
ositive. If det(A) = 0, the matrix is singular, which is a key idea in AQA exam questions.
不要将这种竖线记号与数的绝对值混淆。行列式可以是负数、零或正数。如果 det(A) = 0，则该矩阵是奇异矩阵，这是 AQA 考题中的一个关键概念。
2. Determinant of a 2×2 matrix | 2×2 矩阵的行列式
minant is given by ad − bc. This is one of the most frequently used formulas in the AQA specification.
对于 2×2 矩阵 A = [a b; c d]，其行列式为 ad − bc。这是 AQA 考试大纲中最常用的公式之一。
det(A) = ad − bc
For example, if A = [3 1; 2 5], then det(A) = 3×5 − 1×2 = 15 − 2 = 13.
例如，若 A = [3 1; 2 5]，则 det(A) = 3×5 − 1×2 = 15 − 2 = 13。
n be inverted. If det(A) ≠ 0, an inverse exists; if det(A) = 0, the matrix has no inverse.
这个单一数值能立即告诉你矩阵是否可逆。若 det(A) ≠ 0，则存在逆矩阵；若 det(A) = 0，则矩阵没有逆矩阵。
3. Geometric meaning of the 2×2 determinant | 2×2 行列式的几何意义
near transformation of the plane. The determinant of this matrix gives the area scale factor of the transformation. The unit square, which has area 1, is transformed into a parallelogram whose signed area is det(A).
2×2 矩阵可以解释为平面上的线性变换。该矩阵的行列式给出变换的面积缩放因子。面积为 1 的单位正方形会被变换为一个平行四边形，其带符号面积就是 det(A)。
The absolute value |det(A)| = 0, the transformation collapses the plane onto a line or a point, so area becomes zero.
绝对值 |det(A)| 是实际的面积缩放因子。如果 det(A) 为负，则变换反转方向，通常包含反射。如果 det(A) = 0，则变换将平面压缩到一条直线或一个点上，因此面积变为零。
This geometric interpretation helps you check whether answers are reasonable. An area scale factor cannot be negative when measuring real area, so you often quote the modulus of the determinant in applied geometry questions.
这种几何解释有助于检查答案是否合理。在测量实际面积时，面积缩放因子不能为负数，因此在应用几何题中通常要取行列式的模。
4. Determinant of a 3×3 matrix | 3×3 矩阵的行列式
ng the first row. The formula uses 2×2 minors, each with a sign from the cofactor pattern.
对于 3×3 矩阵 A = [a b c; d e f; g h i]，可以通过沿第一行展开来求行列式。公式使用 2×2 子式，并且每个子式都带有由代数余子式符号规则给出的正负号。
det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
matrix left over when the row and column containing that entry are deleted. The signs follow the pattern positive, negative, positive for the first row.
每一项都是第一行元素与删除该元素所在行和列后剩下的 2×2 矩阵的行列式之积。第一行的符号规律为正、负、正。
The same method can be used by expanding along any row or column, provided you use the correct sign pattern. For AQA exams, expanding along a row or column with zeros often saves time.
同样的方法也可以沿任意行或列展开，只要使用正确的符号规律即可。在 AQA 考试中，沿含有零的行或列展开通常可以节省时间。
5. Sarrus' rule for 3×3 determinants | 3×3 行列式的萨鲁斯法则
Sarrus' only for 3×3 matrices. It cannot be used for 2×2, 4×4 or larger matrices. The rule states:
萨鲁斯法则是一种只适用于 3×3 矩阵的快速计算方法。它不能用于 2×2、4×4 或更大的矩阵。该法则如下：
det(A) = aei + bfg + cdh − afh − bdi − ceg
To remember this, write the first two columns of the matrix again to the right of the original matrix. The three positive terms come from diagonals going down and to the right; the three negative terms come from the other diagonals.
为了记住这个公式，可以把矩阵的前两列重复写在原矩阵的右侧。三个正项来自向右下方延伸的对角线；三个负项来自另一组对角线。
Although Sarrus' rule is useful, you must be careful with signs. Many exam mistakes come from writing the negative terms incorrectly or using Sarrus when it does not apply.
虽然萨鲁斯法则很实用，但必须注意符号。许多考试错误源于负项写错，或在不适用的情况下错误使用萨鲁斯法则。
6. Key properties of determinants | 行列式的主要性质
These properties are important for simplifying calculations and answering proof-style questions. You should know them for AQA Further Mathematics papers.
以下性质对于简化计算和解答证明类题目非常重要。在 AQA 进阶数学考试中应当熟练掌握。
- det(I) = 1 — 单位矩阵的行列式为 1。
- det(Aᵀ) = det(A) — 转置矩阵的行列式与原矩阵相同。
- det(AB) = det(A)det(B) — 两个矩阵乘积的行列式等于各自行列式的乘积。
- det(kA) = kⁿdet(A) — 对于 n×n 矩阵，kA 的行列式等于 kⁿ 乘以 det(A)。
- Swapping two rows changes the sign — 交换两行，行列式变号。
- Two equal rows give det(A) = 0 — 两行完全相同，行列式为零。
- Adding a multiple of one row to another does not change det(A) — 将一行的倍数加到另一行，行列式不变。
- The determinant of a triangular matrix is the product of its diagonal entries — 三角矩阵的行列式等于其对角线元素之积。
sume det(A + B) = det(A) + det(B). This is false, and AQA examiners often test this point.
一个常见误区是认为 det(A + B) = det(A) + det(B)。这是错误的，AQA 考官经常考查这一点。Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图