5 Rules for Manipulating Determinants | 行列式运算的5条规则

📚 5 Rules for Manipulating Determinants | 行列式运算的5条规则

The determinant is a single number calculated from a square matrix. It tells us whether a matrix is invertible and is used throughout algebra, geometry and systems of equations. For A-Level Mathematics, you need to know how to manipulate determinants efficiently using a small set of key rules.

行列式是从方阵计算出的一个数值。它告诉我们矩阵是否可逆，并在代数、几何和方程组中广泛使用。对于A-Level数学，你需要掌握少量关键规则，才能高效地进行行列式运算。

1. What Is a Determinant? | 什么是行列式？

For a 2 × 2 matrix, the determinant is defined as follows:

对于2×2矩阵，行列式的定义如下：

If A = [ a b ; c d ], det(A) = ad − bc

For a 3 × 3 matrix, the determinant can be found by expansion along a row or column, or by applying row operations. In AQA A-Level Mathematics, both 2 × 2 and 3 × 3 determinants are expected.

对于3×3矩阵，可以通过沿某一行或某一列展开来计算行列式，也可以通过行变换来计算。在AQA A-Level数学中，2×2和3×3的行列式都在考查范围内。

2. Rule 1: Transpose Does Not Change the Determinant | 规则1：转置不改变行列式

The transpose of a matrix is obtained by swapping its rows and columns. For any square matrix A, the determinant of Aᵀ is equal to the determinant of A.

矩阵的转置是通过交换行和列得到的。对任意方阵A，Aᵀ的行列式等于A的行列式。

det(Aᵀ) = det(A)

This rule is useful because a statement that works for rows in a determinant also works for columns. You may choose to expand along whichever row or column makes the calculation easier.

这条规则很有用，因为对行列式中的行成立的结论，对列也成立。你可以选择沿计算更方便的那一行或列展开。

3. Rule 2: Swapping Two Rows/Columns Changes Sign | 规则2：交换两行/两列改变符号

If two rows of a matrix are swapped, the determinant changes sign. The same is true if two columns are swapped.

如果交换矩阵的两行，行列式变号。交换两列也是如此。

det after swapping row i and row j = − det(A)

For example, for a 2 × 2 matrix:

例如，对于2×2矩阵：

det[ a b ; c d ] = ad − bc, but det[ c d ; a b ] = bc − ad = −(ad − bc)

This sign change is essential when using row operations to simplify a determinant before evaluating it.

这一变号规则在使用行变换简化行列式时至关重要。

4. Rule 3: Multiplying a Row/Column by a Scalar Multiplies the Determinant | 规则3：某行/列乘以纯量，行列式乘以该纯量

If exactly one row or one column of a matrix is multiplied by a scalar k, the determinant is multiplied by k. This is different from multiplying the whole matrix by k, which multiplies the determinant by kⁿ for an n × n matrix.

如果仅将矩阵的一行或一列乘以纯量k，行列式就乘以k。这与整个矩阵乘以k不同：n×n矩阵整体乘以k，行列式乘以kⁿ。

For an n × n matrix A, det(kA) = kⁿ det(A)

For example, for a 2 × 2 matrix:

例如，对于2×2矩阵：

det[ 2a 2b ; c d ] = 2(ad − bc) = 2 det(A)

This rule allows you to factor out common factors from a row or column before calculating a determinant.

这条规则允许你在计算行列式之前，从某一行或列中提取公因数。

5. Rule 4: Adding a Multiple of One Row/Column to Another Leaves the Determinant Unchanged | 规则4：将一行/列的倍数加到另一行/列，行列式不变

The most powerful row operation is adding a multiple of one row to another row. This operation does not change the determinant.

最有力的行变换是将一行的倍数加到另一行上。这个操作不改变行列式。

Rᵢ → Rᵢ + kRⱼ gives same determinant

The same rule applies to columns:

同样的规则也适用于列：

Cᵢ → Cᵢ + kCⱼ gives same determinant

This is very useful for creating zeros in a matrix before expanding a determinant. For example, to make a zero below a leading entry, you can subtract a suitable multiple of the first row from the second row.

这非常有用，可以在展开行列式前先制造零元素。例如，要使首项下方出现零，可以将第一行的适当倍数从第二行中减去。

6. Rule 5: Determinant of a Product = Product of Determinants | 规则5：矩阵乘积的行列式等于各自行列式的乘积

For two square matrices A and B of the same size, the determinant of their product is the product of their determinants.

对于两个同阶方阵A和B，它们乘积的行列式等于各自行列式的乘积。

det(AB) = det(A) × det(B)

This rule also gives a useful test for inverses. Since det(A A⁻¹) = det(I) = 1, we have det(A) × det(A⁻¹) = 1, so det(A⁻¹) = 1 / det(A) provided det(A) ≠ 0.

这条规则也给出了判断逆矩阵的有用条件。因为det(A A⁻¹) = det(I) = 1，所以det(A) × det(A⁻¹) = 1，于是当det(A) ≠ 0时，det(A⁻¹) = 1 / det(A)。

7. Applying the Rules to Evaluate a 3×3 Determinant | 运用规则计算3×3行列式

You can combine the rules above to simplify a 3 × 3 determinant before expanding. The aim is to create a row or column with as many zeros as possible.

你可以结合以上规则，在展开之前简化3×3行列式。目标是制造尽可能多零元素的行或列。

Consider the determinant:

考虑以下行列式：

| 1 2 3 ; 4 5 6 ; 7 8 9 |

Use the operation R₂ → R₂ − 4R₁ and R₃ → R₃ − 7R₁. These do not change the determinant:

使用行变换R₂ → R₂ − 4R₁和R₃ → R₃ − 7R₁。这些操作不改变行列式：

| 1 2 3 ; 0 −3 −6 ; 0 −6 −12 |

Now use R₃ → R₃ − 2R₂:

再用R₃ → R₃ − 2R₂：

| 1 2 3 ; 0 −3 −6 ; 0 0 0 | = 0

Since the determinant has a zero row, the determinant is 0. The original matrix is singular, meaning it has no inverse.

因为行列式有一行全为零，所以行列式为0。原矩阵是奇异矩阵，意味着它没有逆矩阵。

8. Worked Example | 例题

Evaluate the determinant of the matrix:

计算以下矩阵的行列式：

A = [ 2 1 1 ; 1 3 2 ; 1 0 0 ]

Expand along the third row because it contains two zeros:

沿第三行展开，因为它包含两个零：

det(A) = 1 × det[ 1 1 ; 3 2 ]

Notice the sign pattern for expansion along row 3: positions (3,1) and (3,3) have positive signs, while (3,2) has a negative sign. The position (3,1) contributes +1 × (1×2 − 1×3).

注意沿第3行展开的符号模式：位置(3,1)和(3,3)为正，位置(3,2)为负。位置(3,1)的贡献为+1 × (1×2 − 1×3)。

= 1 × (2 − 3) = −1

So det(A) = −1. Since it is non-zero, matrix A is invertible.

因此det(A) = −1。由于行列式不为零，矩阵A可逆。

9. Common Mistakes | 常见错误

Writing det(kA) = k det(A) for an n × n matrix. The correct result is det(kA) = kⁿ det(A).

错误地写det(kA) = k det(A)。正确的结论是det(kA) = kⁿ det(A)。
Forgetting the sign change when swapping rows or columns.

忘记交换行或列时行列式要变号。
Using Rᵢ → Rᵢ + kRⱼ and also multiplying by k. The row addition does not change the determinant, but scaling a row does.

混淆Rᵢ → Rᵢ + kRⱼ与乘以k。行加法不改变行列式，但缩放一行会改变行列式。
Expanding a 3 × 3 determinant without using the correct alternating sign pattern +, −, +.

展开3×3行列式时没有使用正确的交替符号模式+，−，+。
Confusing transpose with inverse. Transpose does not change the determinant, but inverse is the reciprocal of the determinant.

混淆转置与逆。转置不改变行列式，但逆矩阵的行列式是原行列式的倒数。

10. Exam Tips | 考试提示

In AQA A-Level Mathematics, always state which row or column you are using when expanding a determinant. Show the 2 × 2 determinants clearly after reducing a 3 × 3 determinant.

在AQA A-Level数学考试中，展开行列式时一定要说明你沿用的行或列。将3×3行列式化为2×2行列式后，要清楚写出每一个2×2行列式。

Use the manipulation rules to create zeros. A zero row or zero column immediately tells you the determinant is 0. If a question asks whether a matrix is singular, just check whether its determinant is 0.

使用运算规则制造零元素。一行或一列全为零时，可以直接判断行列式为0。如果题目问矩阵是否奇异，只需判断其行列式是否为0。

For 2 × 2 matrices, remember the formula ad − bc exactly. For 3 × 3 matrices, choose the row or column with the most zeros to reduce arithmetic mistakes.

对于2×2矩阵，要准确记住ad − bc公式。对于3×3矩阵，选择零元素最多的行或列展开，以减少计算错误。

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5 Rules for Manipulating Determinants | 行列式运算的5条规则