📚 The Determinant of a Square Matrix | 方阵的行列式
The determinant is a fundamental concept in linear algebra that assigns a scalar value to every square matrix. It captures essential information about the matrix, such as whether it is invertible, the scale factor of geometric transformations, and the nature of solutions to linear systems. In the IB Mathematics: Analysis and Approaches and Applications and Interpretation courses, a solid understanding of determinants supports work on matrices, vectors, eigenvalues and solving equations.
行列式是线性代数中为每个方阵赋予一个标量值的基本概念。它捕捉了矩阵的关键信息，例如矩阵是否可逆、几何变换的缩放因子以及线性方程组解的性质。在IB数学：分析与方法和应用与解释课程中，扎实理解行列式有助于学习矩阵、向量、特征值以及方程求解。
1. What is a Determinant? | 什么是行列式？
Every square matrix A of size n × n has a unique number associated with it, called the determinant, denoted by det(A) or |A|. It is calculated from the entries of the matrix according to specific rules. The value of a determinant can be positive, negative or zero, and it changes predictably under row operations.
每个 n × n 方阵 A 都有一个唯一的数值与之对应，称为行列式，记作 det(A) 或 |A|。行列式是根据特定规则从矩阵的元素计算出来的。行列式的值可以是正数、负数或零，并且在行变换下会有规律地变化。
A non-zero determinant tells us the matrix is invertible (non-singular), whereas a zero determinant means the matrix is singular and has no inverse. The sign of the determinant also reveals orientation information in geometric settings.
非零行列式表明矩阵可逆（非奇异）；而零行列式意味着矩阵是奇异的，没有逆矩阵。行列式的符号在几何情境下还揭示了方向信息。
The notation |A| uses vertical bars, which look like absolute value signs. This is a common source of confusion: |A| r is. Only square matrices have determinants, so a 2 × 3 matrix has no determinant at all.
记号 |A| 使用竖线，看起来像绝对值符号，这是常见的混淆来源：|A| 可以是负数，而绝对值永远不会是负数。只有方阵才有行列式，因此 2 × 3 矩阵根本没有行列式。
2. Determinant of a 2×2 Matrix | 2×2 矩阵的行列式
For a 2×2 matrix A = [a b; c d], the determinant is given by the straightforward formula below. This can be remembered as the product of the entries on the main diagonal minus the product of the entries on the other diagonal.
对于 2×2 矩阵 A = [a b; c d]，行列式由下面的简单公式给出。可以记作主对角线元素的乘积减去另一条对角线元素的乘积。
det(A) = |A| = ad − bc
Example: If A = [3 2; 1 4] 3 × 4 − 2 × 1 = 12 − 2 = 10. Since the determinant is non-zero, the matrix is invertible.
例：若 A = [3 2; 1 4]，则 det(A) = 3 × 4 − 2 × 1 = 12 − 2 = 10。由于行列式非零，该矩阵可逆。
Example (singular): If B = [2 4; 1 2], then det(B) = 2 × 2 − 4 × 1 = 4 − 4 = 0, so B is singular and has no inverse. Notice that the second row is exactly half the first row, which is the algebraic reason the determinant vanishes.
例（奇异矩阵）：若 B = [2 4; 1 2]，则 det(B) = 2 × 2 − 4 × 1 = 4 − 4 = 0，因此 B 是奇异矩阵，没有逆矩阵。可以看到第二行正好是第一行的一半，这正是行列式为零的代数原因。
Geometrically, |det(A)| is the area scale factor of the linear transformation defined by A. A unit square is mapped to a parallelogram whose area equals |ad − bc| e parallelogram collapses onto a line or a point, so the transformation is not reversible.
从几何上看，|det(A)| 是矩阵 A 所定义的线性变换的面积缩放因子。单位正方形被映射为面积等于 |ad − bc| 的平行四边形。如果 det(A) = 0，该平行四边形塌缩成一条线或一个点，因此该变换不可逆。
3. Determinant of a 3×3 Matrix: Cofactor Expansion | 3×3 矩阵的行列式：余子式展开
For a 3×3 matrix A = [a b c; d e f; g h i] f the 2×2 matrix obtained by deleting its row and column, and the signs alternate +, −, +.
对于 3×3 矩阵 A = [a b c; d e f; g h i]，行列式可通过沿第一行展开求得。第一行的每个元素乘以删除其所在行与列后得到的 2×2 矩阵的行列式，符号依次为 +、−、+。
det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Worked example: Let A = [2 1 3; 0 4 5; 1 0 6] − 5 × 0) − 1(0 × 6 − 5 × 1) + 3(0 × 0 − 4 × 1) = 2(24) − 1(−5) + 3(−4) = 48 + 5 − 12 = 41.
例题：设 A = [2 1 3; 0 4 5; 1 0 6]。沿第一行展开得 − 5 × 0) − 1(0 × 6 − 5 × 1) + 3(0 × 0 − 4 × 1) = 2(24) − 1(−5) + 3(−4) = 48 + 5 − 12 = 41。
The great advantage of cofactor expansion is that you can choose any row or any column. Choosing a row or column containing zeros drastically reduces the work, because the corresponding terms become zero and can be skipped entirely.
余子式展开的最大优势在于可以任选一行或任一列。选择含零较多的行或列会大幅减少计算量，因为相应的项变为零，可以直接省略。
ofactor expansion becomes painfully slow. At that point, row reduction to triangular form or a calculator or computer is the sensible route.
对于更大的矩阵（4×4 及以上），余子式展开会变得非常繁琐。此时，化为三角形矩阵的行化简方法，或使用计算器、计算机才是明智的选择。
4. The Sarrus Rule for 3×3 Matrices | 3×3 矩阵的萨吕斯法则
A quick shortcut for 3×3 determinants is Sarrus' s along the downward diagonals and subtract the three products along the upward diagonals.
3×3 行列式有一个快捷方法叫萨吕斯法则，也称对角线法则。把矩阵的前两列抄到右侧，然后加上三条向下对角线的乘积，再减去三条向上对角线的乘积。
det(A) = aei + bfg + cdh − ceg − bdi − afh
Checking this against the earlier example A = [2 1 3; 0 4 5; 1 0 6] × 5 × 0 = 0, giving 12. Therefore det(A) = 53 − 12 = 41, matching the cofactor expansion.
用前面的例子 A = [2 1 3; 0 4 5; 1 0 6] 验证：向下对角线的乘积为 2 × 4 × 6 = 48、1 × 5 × 1 = 5、3 × 0 × 0 = 0，合计 53。向上对角线的乘积为 3 × 4 × 1 = 12、1 × 0 × 6 = 0、2 × 5 × 0 = 0，合计 12。因此 det(A) = 53 − 12 = 41，与余子式展开结果一致。
Important warning: Sarrus' answers. For 4×4 matrices, use cofactor expansion with a zero-rich row, or row reduction.
重要提醒：萨吕斯法则只适用于 3×3 矩阵。它不能推广到 4×4 或更大的行列式，否则会得到错误答案。处理 4×4 行列式时，应选择含零较多的行进行余子式展开，或使用行化简。
5. Minors and Cofactors | 余子式与代数余子式
The minor Mij of an entry aij is the determinant of the matrix that remains after deleting row i and column j. The cofactor Cij is the minor with a sign attached according to the position of the entry.
元素 aij 的余子式 Mij 是删除第 i 行与第 j 列后剩余矩阵的行列式。代数余子式 Cij 则是在余子式前根据元素位置附加符号。
Cij = (−1)i + j Mij
the pattern is + − + on the first row, − + − on the second row and + − + on the third row.
符号遵循棋盘格式，左上角为 +。对 3×3 矩阵而言，第一行为 + − +，第二行为 − + −，第三行为 + − +。
With cofactors, the determinant can be written as an expansion along any row or column. Expanding along row i or column j gives the same final value, which is a very useful self-checking tool in exams.
利用代数余子式，行列式可以沿任意一行或任意一列展开。沿第 i 行或第 j 列展开得到的最终值相同，这是考试中非常有用的自查手段。
det(A) = Σj aijCij = Σi aijCij
A useful identity is that the sum of the entries of one row multiplied by the cofactors of a different row equals zero, because that combination corresponds to a determinant with two identical rows.
一个有用的恒等式是：某一行元素与另一行对应代数余子式的乘积之和等于零，因为这种组合对应于一个有两行相同的行列式。
6. Key Properties of Determinants | 行列式的重要性质
Determinants obey a set of algebraic rules that make calculation far easier. The most important ones are collected below, and they are frequently tested in IB examination questions.
行列式遵循一组代数法则，使计算大大简化。下面列出最重要的几条，它们在 IB 考试中经常被考查。
| Property | 性质 | Statement | 表述 |
| Product rule | 乘积法则 | det(AB) = det(A) × det(B) |
| Transpose | 转置 | det(AT) = det(A) |
| Scalar multiple | 数乘 | det(kA) = kn det(A) for an n × n matrix |
| Inverse | 逆矩阵 | det(A−1) = 1 / det(A) |
| Triangular matrix | 三角形矩阵 | det = product of diagonal entries |
| Identity | 单位矩阵 | det(I) = 1 |
= 2³ det(A) = 8 det(A), not 2 det(A), because every one of the n rows is multiplied by k.
数乘法则最容易出错。对于 3×3 矩阵，det(2A) = 2³ det(A) = 8 det(A)，而不是 2 det(A)，因为 n 行中的每一行都被乘以 k。
Two further facts are extremely useful. First, if two rows (or two columns) of a matrix are identical, or if one row is a scalar multiple of another, the determinant is zero. Second, if any row or column consists entirely of zeros, the determinant is zero. Both facts follow directly from the geometry of the linear transformation.
另外两条事实非常有用。第一，若矩阵的两行（或两列）完全相同，或其中一行是另一行的倍数，则行列式为零。第二，若任意一行或一列全为零，则行列式为零。这两条都可以直接从线性变换的几何意义推出。
7. Effect of Row Operations | 行变换对行列式的影响
Because a determinant is built from the rows of a matrix, elementary row operations change it in three precise ways. Knowing these rules lets you turn any matrix into triangular form and read the determinant off the diagonal.
由于行列式由矩阵的各行构造而成，初等行变换会以三种明确的方式改变它。掌握这些规则，就能把任意矩阵化为三角形矩阵，并直接从对角线读出行列式。
- Row swap: swapping two rows multiplies the determinant by −1.
- Row scaling: multiplying a row by a non-zero constant k multiplies the determinant by k.
- Row addition: adding a multiple of one row to another row leaves the determinant unchanged.
- 交换两行：行列式乘以 −1。
- 行乘常数：将某一行乘以非零常数 k，行列式乘以 k。
- 行相加：将某一行的倍数加到另一行，行列式不变。
Row addition is the workhorse of computation because it is determinant-preserving. Row swapping is also used to bring a non-zero entry into the pivot position, but you must remember to record the sign change.
行相加是计算的主力，因为它不改变行列式。交换两行也常用来把非零元素调到主元位置，但必须记住记录符号变化。
Example: Starting from A = [2 1 3; 0 4 5; 1 0 6] 6. However, this is not det(A), because we subtracted from a row using a fraction derived from row 1 — check is the direct expansion giving 41. Careful tracking of each operation is essential.
例：从 A = [2 1 3; 0 4 5; 1 0 6] 出发，做 R₃ → R₃ − ½R₁，得到上三角矩阵 [2 1 3; 0 4 5; 0 −0.5 4.5]，其行列式为 2 × 4 × 4.5 = 36。这个结果显示：精确记录每一步操作至关重要，而直接展开得到的 41 始终可以作为验证标准。
A practical strategy in exams is: reduce to triangular form using only row addition and row swapping, then multiply the diagonal entries, correcting for the number of swaps performed.
考试中的实用策略是：只用行相加与行交换化为三角形矩阵，然后乘以对角线元素，并根据交换次数修正符号。
8. Determinants and Invertibility | 行列式与可逆性
The single most important interpretation of a determinant is as an invertibility test. A square matrix A if and only if det(A) ≠ 0. When det(A) = 0 the matrix is singular and cannot be inverted.
行列式最重要的一个解释就是可逆性判据。方阵 A 有逆矩阵当且仅当 det(A) ≠ 0。当 det(A) = 0 时，矩阵是奇异的，不可求逆。
itten down explicitly using the determinant and the adjugate. Notice how the diagonal entries swap positions and the off-diagonal entries change sign.
对于 2×2 矩阵，可以用行列式与伴随矩阵直接写出逆矩阵。注意对角线元素互换位置，非对角线元素改变符号。
A−1 = 1 / (ad − bc) × [d −b; −c a]
For larger matrices the same idea generalises: A−1 = adj(A) / det(A) 3×3 inverses, but you should still be able to justify invertibility using the determinant.
对更大的矩阵，同样的思想可以推广：A−1 = adj(A) / det(A)，其中 adj(A) 是代数余子式矩阵的转置。在 IB 的解题实践中，3×3 逆矩阵通常用图形计算器完成，但你仍应能用行列式说明可逆性。
A useful consistency check: for the 2×2 case, compute A × A−1 0; 0 1]. If det(A) = 0 the calculation breaks down because you would be dividing by zero.
一个实用的检验方法：在 2×2 情形下计算 A × A−1，确认得到单位矩阵 I = [1 0; 0 1]。如果 det(A) = 0，计算就失效了，因为相当于除以零。
9. Cramer's Rule and Linear Systems | 克拉默法则与线性方程组
Determinants give a neat closed-form method for solving linear systems. For a system written in matrix form Ax = b, if det(A) ≠ 0 then the unique solution is found by replacing the appropriate column with the vector b and dividing by det(A).
行列式为求解线性方程组提供了一种简洁的闭式方法。对于写成矩阵形式 Ax = b 的方程组，若 det(A) ≠ 0，则唯一解可通过用向量 b 替换相应的列，再除以 det(A) 得到。
x = Dx / D, y = Dy / D, z = Dz / D
Here D = det(A), while Dx, Dy and Dz are the determinants obtained by replacing the x, y or z column of A with the constant column b.
这里 D = det(A)，而 Dx、Dy、Dz 分别是把 A 中 x、y 或 z 对应的列替换为常数向量 b 后得到的行列式。
Cramer's rule also explains the geometry of systems. If D ≠ 0 the system has exactly one solution. If D = 0 but at least one of Dx, Dy is non-zero, there is no solution. If all of the determinants are zero, the system has infinitely many solutions or none at all, and further analysis is needed.
克拉默法则也解释了方程组的几何结构
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