Properties of Determinants | 行列式的性质

📚 Properties of Determinants | 行列式的性质

Understanding the properties of determinants is crucial for solving linear equations, finding inverses, and grasping advanced topics in linear algebra. This article outlines the fundamental properties every IB Mathematics student should master, with clear explanations and examples.

理解行列式的性质对于解线性方程组、求逆矩阵以及掌握线性代数的高级主题至关重要。本文概述了每位 IB 数学学生应掌握的基本性质，并配有清晰的解释和示例。

1. Determinant of a Transpose | 转置行列式

The determinant of a matrix and its transpose are equal. For any square matrix A, det(A^T) = det(A).

矩阵的行列式与其转置的行列式相等。对于任意方阵 A，det(A^T) = det(A)。

This property implies that any statement about rows of a determinant also holds true for columns, effectively doubling the applicability of all row-based properties.

这一性质表明，关于行列式行的任何陈述对列也同样成立，从而将所有基于行的性质的适用范围有效地扩大了一倍。

det(A^T) = det(A)

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2. Row Interchange Property | 行交换性质

changes. In other words, det(B) = −det(A) where B is obtained from A by swapping two rows.

如果将行列式中的两行（或两列）互换，行列式的符号会改变。换言之，若矩阵 B 由 A 交换两行得到，则 det(B) = −det(A)。

For a 2×2 matrix, swapping rows gives:
det([c d; a b]) = cb − da = −(ad − bc) = −det([a b; c d]).

以 2×2 矩阵为例，交换行后：
det([c d; a b]) = cb − da = −(ad − bc) = −det([a b; c d])。

det(A with swapped rows) = −det(A)

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3. Multiplying a Row by a Scalar | 行乘以标量

That is, if matrix B is formed by multiplying one row of A by k, then det(B) = k · det(A).

如果行列式某一行（或一列）的所有元素都乘以标量 k，则整个行列式的值也乘以 k。即若矩阵 B 由 A 的某行乘以 k 得到，则 det(B) = k · det(A)。

For example, multiplying the first row of a 2×2 matrix by k:
det([ka kb; c d]) = (ka)d − (kb)c = k(ad − bc) = k det(A).

例如，将 2×2 矩阵的第一行乘以 k：
det([ka kb; c d]) = (ka)d − (kb)c = k(ad − bc) = k det(A)。

det(k × rowᵢ) = k · det(A)

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4. Row with All Zeros | 零行性质

If a square matrix has a row or a column consisting entirely of zeros, its determinant is zero. This follows directly from the definition of determinant as a sum of products, each involving one element from every row.

如果方阵的某一行或某一列全为零，则其行列式为零。这直接源于行列式的定义，即每个乘积项都必须包含每一行的一个元素，因此零行使得每个乘积都为零。

([0 0; c d]) = 0·d − 0·c = 0. Similarly, any larger matrix with a zero row yields det = 0.

det([0 0; c d]) = 0·d − 0·c = 0。同理，任何包含零行的大矩阵都会使行列式为零。

If any row = [0 0 … 0], then det = 0

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5. Proportional Rows (or Columns) | 成比例的行或列

If two rows (or columns) of a matrix are proportional, i.e., one is a scalar multiple of the other, the determinant is zero. This is because subtracting the appropriate multiple of one row from the other creates a zero row without changing the determinant.

如果矩阵的两行（或两列）成比例，即一行是另一行的标量倍数，则行列式为零。原因是通过从一行减去另一行的适当倍数可以产生一个零行，而不改变行列式的值。

w₂ = k · row₁, then det = 0. A common special case is when two rows are identical (k = 1).

例如，若 row₂ = k · row₁，则 det = 0。一个常见的特例是两行完全相同（k = 1）。

det(A with two proportional rows) = 0

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6. Additivity in a Single Row | 单行可加性

If one row of a matrix can be written as the sum of two row vectors, the determinant can be expressed as the sum of two determinants, each containing one of those row vectors in the same position, with all other rows unchanged.

如果矩阵的某一行可以写成两个行向量之和，则行列式可表示为两个行列式之和，这两个行列式分别在相同位置包含其中一个行向量，其余行保持不变。

For a 2×2 matrix, if the first row is [x₁+y₁, x₂+y₂]:
det([x₁+y₁ x₂+y₂; c d]) = det([x₁ x₂; c d]) + det([y₁ y₂; c d]).

以 2×2 矩阵为例，如果第一行是 [x₁+y₁, x₂+y₂]：
det([x₁+y₁ x₂+y₂; c d]) = det([x₁ x₂; c d]) + det([y₁ y₂; c d])。

This property reflects the multilinear nature of the determinant.

这一性质反映了行列式的多重线性特征。

det(A with rowᵢ = v + w) = det(A with v) + det(A with w)

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7. Row Addition Property | 行加法性质

Adding a multiple of one row to another row does not change the value of the determinant. For example, replacing rowᵢ with rowᵢ + k·rowⱼ leaves the determinant unchanged. This is the key principle behind Gaussian elimination used to compute determinants.

将一行的倍数加到另一行上不会改变行列式的值。例如，将第 i 行替换为 rowᵢ + k·rowⱼ，行列式不变。这是高斯消元法计算行列式背后的关键原理。

Proof: det(rowᵢ + k·rowⱼ) = det(rowᵢ) + k·det(rowⱼ, same row as rowⱼ ivity and scalar multiplication. Since the second term has two identical rows, it is zero.

证明：由可加性和标量乘法，det(rowᵢ + k·rowⱼ) = det(rowᵢ) + k·det(出现两个相同行的情况)。因为第二项包含两行相同，故为零。

det(A after rowᵢ → rowᵢ + k·rowⱼ) = det(A)

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8. Determinant of a Triangular Matrix | 三角形矩阵的行列式

The determinant of an upper triangular, lower triangular, or diagonal matrix is simply the product of its diagonal entries. This makes calculation extremely efficient once a matrix is transformed into triangular form.

上三角矩阵、下三角矩阵或对角矩阵的行列式仅为其对角线元素的乘积。一旦矩阵被转化为三角形形式，计算就变得极其高效。

For a 3×3 upper triangular matrix with diagonal elements a, e, i:
det = a × e × i. Similarly, det = a₁₁ × a₂₂ × … × aₙₙ for n×n.

对于对角线元素为 a, e, i 的 3×3 上三角矩阵：
det = a × e × i。类似地，对于 n×n 矩阵，det = a₁₁ × a₂₂ × … × aₙₙ。

det(triangular) = a₁₁ · a₂₂ · … · aₙₙ

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9. Determinant of a Product | 乘积的行列式

. This property is fundamental for understanding matrix invertibility and transformations.

对于任意两个同阶方阵 A 和 B，乘积的行列式等于行列式的乘积：det(AB) = det(A) · det(B)。该性质对于理解矩阵可逆性和变换至关重要。

An important consequence is that det(A^k) = (det(A))^k for any positive integer k. Also, if AB is invertible, then both A and B must be invertible.

一个重要的推论是：对于任意正整数 k，det(A^k) = (det(A))^k。而且，如果 AB 可逆，则 A 和 B 都必须可逆。

det(AB) = det(A) · det(B)

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10. Determinant and Inverse | 行列式与逆矩阵

A square matrix A is invertible if and only if its determinant is non-zero. The determinant of the inverse matrix is the reciprocal of the original determinant: det(A⁻¹) = 1/det(A).

方阵 A 可逆当且仅当其行列式不为零。逆矩阵的行列式是原行列式的倒数：det(A⁻¹) = 1/det(A)。

This follows from det(A·A⁻¹) = det(I) = 1, and det(A·A⁻¹) = det(A) · det(A⁻¹). Thus, a non-zero determinant guarantees the existence of a unique inverse.

这是因为 det(A·A⁻¹) = det(I) = 1，而 det(A·A⁻¹) = det(A) · det(A⁻¹)。因此，非零行列式保证了唯一逆矩阵的存在。

det(A⁻¹) = 1 / det(A), (det(A) ≠ 0)

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11. Determinant and Eigenvalues | 行列式与特征值

nvalues, counted with algebraic multiplicity. For an n×n matrix A with eigenvalues λ₁, λ₂, …, λₙ, we have det(A) = λ₁ · λ₂ · … · λₙ.

矩阵的行列式等于其特征值的乘积（计代数重数）。对于特征值为 λ₁, λ₂, …, λₙ 的 n×n 矩阵 A，有 det(A) = λ₁ · λ₂ · … · λₙ。

This connection is particularly useful for inferring the determinant from known eigenvalues, or for concluding that a matrix is singular if at least one eigenvalue is zero.

这一联系对于从已知特征值推断行列式，或根据至少一个特征值为零来断定矩阵是奇异的，特别有用。

For a diagonalizable matrix, this can be seen by diagonalizing: det(PDP⁻¹) = det(D) = product of diagonal entries (eigenvalues).

对于可对角化的矩阵，可通过对角化看出：det(PDP⁻¹) = det(D) = 对角线元素（特征值）的乘积。

det(A) = λ₁ · λ₂ · … · λₙ

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12. Summary of Key Properties | 关键性质总结

The following table recaps the main determinant properties for quick reference. Understanding and applying these will greatly simplify matrix computations in IB Mathematics.

下表总结了主要的行列式性质，方便快速查阅。理解并应用这些性质将极大简化 IB 数学中的矩阵计算。

Property	Expression
Transpose	det(AT) = det(A)
Row swap	det(B) = −det(A)
Row multiplication by k	det = k · det(A)
Zero row	det = 0
Proportional rows	det = 0
Row additivity	det(rowᵢ = v+w) = det(Av) + det(Aw)
Row addition (elimination)	det unchanged
Triangular matrix	det = a₁₁·a₂₂·…·aₙₙ
Product	det(AB) = det(A)det(B)
Inverse	det(A−1) = 1/det(A)
Eigenvalues	det(A) = Π λᵢ

By mastering these properties, students can efficiently evaluate determinants, prove linear independence, and understand the geometry of linear transformations.

通过掌握这些性质，学生能够高效地求出行列式的值，证明线性无关性，并理解线性变换的几何意义。

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