Determinants | 行列式

📚 Determinants | 行列式

The determinant is a special number that can be calculated from a square matrix. In A-level mathematics, it tells us whether a matrix has an inverse, how transformations scale areas or volumes, and whether a system of simultaneous equations has a unique solution.

行列式是一个由方阵计算出的特殊数值。在 A-level 数学中，它告诉我们一个矩阵是否存在逆矩阵、矩阵变换如何缩放面积或体积，以及一个线性方程组是否有唯一解。

1. What is a determinant? | 什么是行列式？

A determinant is a single real number assigned to a square matrix. For a 1×1 matrix, the determinant is simply the entry itself. For larger square matrices, the determinant is obtained by a carefully defined expansion.

行列式是赋予一个方阵的唯一实数。对于 1×1 矩阵，行列式就是其唯一元素本身。对于更大的方阵，行列式通过精心定义的展开方式计算。

We write the determinant of matrix A as det(A) or |A|. Note that |A| does not mean absolute value; it is the determinant notation used in matrix algebra.

我们把矩阵 A 的行列式记作 det(A) 或 |A|。请注意，这里的 |A| 并不表示绝对值，而是矩阵代数中的行列式记号。

For example, if A = [5], then det(A) = 5. For a 2×2 or 3×3 matrix, the determinant can be expanded by a formula or by using the rule of Sarrus for 3×3 matrices.

例如，若 A = [5]，则 det(A) = 5。对于 2×2 或 3×3 矩阵，行列式可以通过公式计算，3×3 矩阵也可以使用萨鲁斯法则。

2. The determinant of a 2×2 matrix | 二阶矩阵的行列式

For a 2×2 matrix:

对于二阶矩阵：

A = [ a b ; c d ]

The determinant is defined as:

行列式定义为：

det(A) = ad − bc

This is the product of the leading diagonal entries minus the product of the other diagonal entries.

即主对角线元素乘积减去副对角线元素乘积。

Example: A = [ 3 4 ; 2 5 ], det(A) = 3 × 5 − 4 × 2 = 15 − 8 = 7

Since the determinant is nonzero, the matrix is invertible.

由于行列式非零，该矩阵可逆。

3. The determinant of a 3×3 matrix | 三阶矩阵的行列式

For a 3×3 matrix:

对于三阶矩阵：

A = [ a b c ; d e f ; g h i ]

The determinant can be found by expansion along the first row:

行列式可以沿第一行展开得到：

det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)

Each entry in the first row multiplies the determinant of the 2×2 matrix left after deleting its row and column. Signs alternate: plus, minus, plus.

第一行每一项都乘以删除其所在行和列后剩下的 2×2 矩阵的行列式。符号依次为加、减、加。

Example: A = [ 1 2 3 ; 0 1 4 ; 5 6 0 ]

Expanding along the first row:

沿第一行展开：

det(A) = 1(1×0 − 4×6) − 2(0×0 − 4×5) + 3(0×6 − 1×5)

= (0 − 24) − 2(0 − 20) + 3(0 − 5) = −24 + 40 − 15 = 1

Because the determinant is 1, this matrix is invertible and preserves volume exactly.

因为行列式为 1，所以该矩阵可逆，并且精确保持体积不变。

4. Properties of determinants | 行列式的性质

Several useful properties help simplify calculations and solve problems.

以下性质有助于简化计算和解决问题。

An n×n zero matrix has determinant 0. All entries are zero, so every product in the expansion is zero.

n×n 零矩阵的行列式为 0。所有元素均为零，因此展开式中每一项都是零。
The identity matrix has determinant 1. This is consistent with the identity transformation doing nothing.

单位矩阵的行列式为 1。这与恒等变换不改变任何形状的事实一致。
If any two rows (or columns) are identical, the determinant is 0.

如果任意两行（或两列）相同，行列式为 0。
Adding a multiple of one row to another row does not change the determinant. This is often used in row-reduction.

将某行的倍数加到另一行，不改变行列式的值。这一性质常用于行化简。
Swapping two rows changes the sign of the determinant.

交换两行会改变行列式的符号。
det(AB) = det(A) × det(B). The determinant of a product equals the product of the determinants.

det(AB) = det(A) × det(B)。矩阵乘积的行列式等于各自行列式的乘积。

A common exam question asks you to find det(AB) directly from det(A) and det(B). Be careful to multiply, not add.

常见考题要求你由 det(A) 和 det(B) 直接求 det(AB)。注意是相乘而不是相加。

5. Geometric meaning: area and volume scale factor | 几何意义：面积与体积缩放因子

A matrix acting on a shape as a linear transformation changes its size. The absolute value of the determinant gives the scale factor for the area or volume.

矩阵作为线性变换作用于形状会改变其大小。行列式的绝对值给出了面积或体积的缩放因子。

For a 2×2 matrix: the absolute value of the determinant is the area scale factor.

对于二阶矩阵：行列式的绝对值是面积缩放因子。

|det(A)| = area of image of unit square = new area ÷ original area

For a 3×3 matrix: the absolute value of the determinant is the volume scale factor.

对于三阶矩阵：行列式的绝对值是体积缩放因子。

If a triangle has area 5 and is transformed by a matrix with det(A) = −2, then its image has area 5 × |−2| = 10. The negative sign means the orientation of the shape is reversed.

如果某三角形面积为 5，被 det(A) = −2 的矩阵变换，则其像的面积为 5 × |−2| = 10。负号表示形状的定向发生了反转。

This interpretation is important in AQA questions with diagrams: they may give the unit square or a known shape and ask for the area after transformation.

这种解释在 AQA 与图形结合的考题中非常重要：题目可能给出单位正方形或已知形状，要求求变换后的面积。

6. Determinants and inverse matrices | 行列式与逆矩阵

An inverse matrix exists if and only if the determinant is nonzero. For a 2×2 matrix:

逆矩阵存在当且仅当行列式非零。对于二阶矩阵：

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

The factor 1/(ad − bc) multiplies every entry of the matrix. If ad − bc = 0, the inverse is undefined.

因子 1/(ad − bc) 要乘到矩阵的每个元素上。如果 ad − bc = 0，则逆矩阵不存在。

For a 3×3 matrix, the inverse is given by:

对于三阶矩阵，逆矩阵由下式给出：

A⁻¹ = 1/det(A) × adj(A)

where adj(A) is the adjugate matrix formed from cofactors. In an AQA exam, you are usually asked to compute 2×2 inverses, while 3×3 inverses may be assessed in further mathematics contexts.

其中 adj(A) 是由代数余子式构成的伴随矩阵。在 AQA 考试中，通常要求计算二阶逆矩阵，而三阶逆矩阵可能在进阶数学中考到。

Check your answer by verifying A × A⁻¹ = I.

可以通过验证 A × A⁻¹ = I 来检查答案是否正确。

7. Singular matrices and rank | 奇异矩阵与秩

A matrix with determinant 0 is called singular. A singular matrix has no inverse.

行列式为 0 的矩阵称为奇异矩阵。奇异矩阵没有逆矩阵。

For a 2×2 matrix, det(A) = 0 means that one row is a multiple of the other. In terms of transformations, the image flattens into a line, so the area becomes 0.

对二阶矩阵而言，det(A) = 0 意味着其中一行是另一行的倍数。从变换角度看，像被压成一条直线，因此面积为 0。

For a 3×3 matrix, det(A) = 0 means the rank is less than 3. The transformation collapses 3D space onto a plane, a line, or a point.

对三阶矩阵而言，det(A) = 0 意味着秩小于 3。该变换把三维空间压缩到平面、直线或点。

If the determinant is nonzero, the matrix is said to be non-singular or invertible, and the rank equals the order of the matrix.

如果行列式非零，则矩阵称为非奇异矩阵或可逆矩阵，其秩等于矩阵的阶数。

8. Using determinants to solve systems of equations | 利用行列式求解方程组

For a system written as A x = b, where A is a square matrix and x is the vector of unknowns:

对于写成 A x = b 的方程组，其中 A 是方阵，x 是未知数向量：

If det(A) ≠ 0, there is exactly one unique solution: x = A⁻¹ b.

若 det(A) ≠ 0，方程组有唯一解：x = A⁻¹ b。
If det(A) = 0, there is either no solution or infinitely many solutions. You need further investigation using row operations or comparing equations.

若 det(A) = 0，方程组要么无解，要么有无穷多解。你需要通过行变换或比较方程进一步判断。

For two equations in two unknowns, det(A) = 0 means the lines represented by the equations are parallel (no solution) or coincide (infinitely many solutions).

对于两个二元一次方程，det(A) = 0 意味着对应两条直线平行（无解）或重合（无穷多解）。

In three equations with three unknowns, det(A) = 0 suggests the planes may be parallel, coincident, or intersect in a line or not at all.

对于三元一次方程组，det(A) = 0 提示三个平面可能平行、重合、交于一条直线或根本没有公共交点。

Thus, the determinant is a quick test to distinguish a unique solution from a singular case.

因此，行列式是判断唯一解与奇异情形的快速工具。

9. Worked examples | 典型例题

Example 1: Find det(A) for A = [ 2 −1 ; 3 4 ].

例 1：求 A = [ 2 −1 ; 3 4 ] 的 det(A)。

det(A) = 2 × 4 − (−1) × 3 = 8 + 3 = 11

Since det(A) = 11 ≠ 0, A is invertible.

因为 det(A) = 11 ≠ 0，所以 A 可逆。

Example 2: A linear transformation maps a square of area 4 to a parallelogram whose area is 12. Find the absolute value of the determinant of the transformation matrix.

例 2：某线性变换把面积为 4 的正方形映射为面积为 12 的平行四边形。求变换矩阵行列式的绝对值。

|det| = 12 ÷ 4 = 3

Example 3: For B = [ 1 2 ; 3 k ], find the value of k for which B is singular.

例 3：对于 B = [ 1 2 ; 3 k ]，求使 B 为奇异矩阵的 k 值。

Set det(B) = 0:

令 det(B) = 0：

1 × k − 2 × 3 = 0 ⇒ k = 6

Therefore B is singular when k = 6.

因此当 k = 6 时 B 是奇异矩阵。

10. Common pitfalls and exam tips | 常见错误与考试技巧

Students often make sign errors in the 3×3 expansion. Use a clear pattern for the signs: + − + along the first row.

学生在三阶展开中常犯符号错误。建议使用清晰的符号模式：沿第一行为 + − +。

Do not think |A| means absolute value. The determinant can be negative. Use absolute value only when finding area or volume.

不要把 |A| 理解为绝对值。行列式可以为负。只有在求面积或体积时才取绝对值。
Remember to subtract bc in a 2×2 determinant. The expression ad − bc is not the same as ad + bc.

二阶行列式切记要减去 bc。表达式是 ad − bc，不是 ad + bc。
In the 3×3 expansion, multiply by the correct minor and keep the alternating signs. For entry b in the first row, the sign is minus.

在三阶展开中，要乘以正确的余子式并保持正负交替。第一行的 b 对应的是负号。
Check whether a matrix is singular before trying to find an inverse. If det = 0, stop and state that the inverse does not exist.

求逆矩阵之前先判断矩阵是否奇异。如果 det = 0，直接说明逆矩阵不存在。
When asked for area after transformation, take the absolute value of the determinant. A negative determinant still gives a positive area.

当要求变换后的面积时，取行列式的绝对值。负的行列式仍然给出正的面积。

In AQA exams, show all expansion steps clearly. Partial credit is often awarded for correct minors even if the final value is wrong.

在 AQA 考试中，请清晰展示所有展开步骤。即使最终数值错误，正确的余子式往往也能得到部分分数。

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Determinants | 行列式