Laws of Matrix Arithmetic | 矩阵算术定律

📚 Laws of Matrix Arithmetic | 矩阵算术定律

Matrix arithmetic follows a clear set of rules. Some of these rules are the same as ordinary number arithmetic, but others are surprisingly different. Understanding these laws is essential for solving AQA A-Level Mathematics problems involving matrices.

矩阵运算遵循一套明确的规则。其中一些规则与普通数的运算相同，但另一些却截然不同。理解这些定律对于解决 AQA A-Level 数学中涉及矩阵的问题至关重要。

---

1. Matrix Addition and Scalar Multiplication | 矩阵加法与数乘

ows and columns). The sum is found by adding corresponding entries. For example, if A = (a_ij) and B = (b_ij), then (A + B)_ij = a_ij + b_ij.

两个矩阵相加的前提是它们具有相同的阶数（即行数和列数分别相同）。和矩阵由对应元素相加得到。例如，若 A = (a_ij) 和 B = (b_ij)，则 (A + B)_ij = a_ij + b_ij。

Scalar multiplication means multiplying every entry of a matrix by the same number. If k is a scalar, then (kA)_ij = k a_ij.

数乘是指用同一个数乘以矩阵中的每一个元素。若 k 是标量，则 (kA)_ij = k a_ij。

---

2. Commutativity of Addition | 加法交换律

is holds because ordinary addition of numbers is commutative for each corresponding entry.

矩阵加法满足交换律。对于任意两个同阶矩阵 A 和 B，都有 A + B = B + A。这是因为数的普通加法对每个对应元素都满足交换律。

A + B = B + A

tiplication also commutes with itself: kA = Ak (though the latter is not usually written).

数乘也满足交换性：kA = Ak（不过后者通常不这样书写）。

---

3. Associativity of Addition and Scalar Multiplication | 加法与数乘的结合律

, (A + B) + C = A + (B + C). This allows us to omit brackets when adding several matrices.

矩阵加法满足结合律。对于任意三个同阶矩阵 A、B 和 C，都有 (A + B) + C = A + (B + C)。这使我们可以在对多个矩阵求和时省略括号。

plication is also associative with respect to scalars: for scalars p and q, p(qA) = (pq)A.

数乘对标量也满足结合律：对于标量 p 和 q，p(qA) = (pq)A。

(A + B) + C = A + (B + C), p(qA) = (pq)A

---

4. Matrix Multiplication Definition | 矩阵乘法定义

If A is an m × n matrix and B is an n × p matrix, then the product AB is an m × p matrix.

矩阵乘法仅在第一个矩阵的列数等于第二个矩阵的行数时才有定义。若 A 是 m × n 矩阵，B 是 n × p 矩阵，则乘积 AB 是 m × p 矩阵。

The entry in row i and column j of AB is obtained by multiplying the i-th row of A by the j-th column of B and summing the products:

AB 的第 i 行第 j 列元素通过将 A 的第 i 行与 B 的第 j 列对应相乘再求和得到：

(AB)_ij = Σₖ a_ikb_kj

This is often called the "row-by-column" rule.

这通常称为“行乘列”法则。

---

5. Non-Commutativity of Multiplication | 乘法非交换性

Matrix multiplication is generally not commutative. In other words, AB and BA are usually different. This is a major difference from ordinary number multiplication.

矩阵乘法通常不满足交换律。也就是说，AB 和 BA 通常不相同。这是与普通数乘法的一个重要区别。

4] and B = [5 6; 7 8]. Then AB = [19 22; 43 50] but BA = [23 34; 31 46]. Clearly AB ≠ BA.

例如，设 A = [1 2; 3 4]，B = [5 6; 7 8]。则 AB = [19 22; 43 50]，而 BA = [23 34; 31 46]。显然 AB ≠ BA。

AB ≠ BA in general

Even when both products are defined, their orders may be different unless the matrices are square. Always check whether the product makes sense before calculating it.

即使在两个乘积都有定义的情况下，除非矩阵是方阵，否则它们的阶数也可能不同。计算前务必检查乘积是否有意义。

---

6. Associativity of Multiplication | 乘法结合律

BC). This is a crucial algebraic property because it allows us to group matrix products in any convenient way.

矩阵乘法满足结合律。对于任意在乘积有定义的矩阵 A、B 和 C，都有 (AB)C = A(BC)。这是一个关键性质，它允许我们以任意方便的方式对矩阵乘积进行分组。

(AB)C = A(BC)

This law is used repeatedly when simplifying expressions such as A(BC) or when computing powers of square matrices.

该定律在化简 A(BC) 等表达式或计算方阵的幂时会被反复使用。

---

7. Distributive Laws | 分配律

Matrix multiplication distributes over addition. For matrices of appropriate orders,

矩阵乘法对加法满足分配律。对于阶数合适的矩阵，有：

A(B + C) = AB + AC

and also

以及

(A + B)C = AC + BC

These two laws allow us to expand and factorise matrix expressions, just as we do with numbers, but always keeping the order of multiplication unchanged.

这两条定律允许我们像处理数一样展开和因式分解矩阵表达式，但必须始终保持乘法的顺序不变。

---

广告 / Sponsored

8. The Zero Matrix | 零矩阵

cts as the additive identity: for any matrix A of the same order, A + O = A and O + A = A.

零矩阵是所有元素都为 0 的矩阵，用 O 或 0 表示。零矩阵扮演加法单位元的角色：对任意同阶矩阵 A，有 A + O = A 和 O + A = A。

has an additive inverse -A such that A + (-A) = O. This is used when subtracting matrices.

在加法中，每个矩阵 A 都有加法逆元 -A，使得 A + (-A) = O。这一性质用于矩阵的减法。

roperty: AB = O does not necessarily imply A = O or B = O. This is different from numbers.

然而，在乘法中零矩阵有一个特殊性质：AB = O 不一定意味着 A = O 或 B = O。这与数的情形不同。

---

9. The Identity Matrix | 单位矩阵

The identity matrix, usually denoted I or I_n main diagonal and 0s elsewhere. For example, the 2 × 2 identity matrix is I = [1 0; 0 1].

单位矩阵通常用 I 或 I_n 表示，它是主对角线上的元素为 1、其余元素为 0 的方阵。例如，2 × 2 单位矩阵为 I = [1 0; 0 1]。

The identity matrix is the multiplicative identity. If A is an m × n matrix, then I_mA = A and AI_n = A. In particular, for a square matrix A of order n, AI = IA = A.

单位矩阵是乘法单位元。若 A 是 m × n 矩阵，则 I_mA = A 且 AI_n = A。特别地，对于 n 阶方阵 A，有 AI = IA = A。

AI = IA = A

---

10. Scalar Multiplication and Matrix Powers | 数乘与矩阵幂

aw: for any scalar k and matrices A and B (with compatible orders), k(AB) = (kA)B = A(kB).

数乘与矩阵乘法之间有一个简单规律：对任意标量 k 以及阶数相容的矩阵 A 和 B，有 k(AB) = (kA)B = A(kB)。

powers are defined recursively: A² = AA, A³ = A²A, and so on. The usual index law holds: A^mAⁿ = A^m+n and (A^m)ⁿ = A^mn, because matrix multiplication is associative.

对于方阵，幂可以递归定义：A² = AA，A³ = A²A，以此类推。通常的指数运算律成立：A^mAⁿ = A^m+n，且 (A^m)ⁿ = A^mn，因为矩阵乘法满足结合律。

By convention, A⁰ = I for a non-zero square matrix A.

按照惯例，对于非零方阵 A，约定 A⁰ = I。

---

11. How These Laws Apply to Simultaneous Equations | 这些定律在联立方程中的应用

rix, then the solution is X = A⁻¹B, provided the laws of arithmetic are applied correctly.

在 AQA A-Level 数学中，矩阵常用于求解联立线性方程组。若 AX = B，其中 A 是可逆方阵，则解为 X = A⁻¹B，前提是正确运用算术定律。

ply by A⁻¹ on the left consistently. We cannot write X = BA⁻¹ unless we start from XA = B.

由于矩阵乘法不满足交换律，我们必须在等式两边一致地左乘 A⁻¹。除非从 XA = B 出发，否则不能写成 X = BA⁻¹。

This shows why the non-commutativity law is more than just a theoretical curiosity: it affects practical problem solving.

这表明非交换律不仅仅是一个理论上的趣事：它直接影响实际解题过程。

---

12. Common Pitfalls to Avoid | 常见易错点

Here is a summary of common mistakes students make with matrix arithmetic:

以下列出了学生在矩阵运算中常见的错误：

• Adding matrices of different orders. Addition requires the same order.
• Multiplying matrices whose inner dimensions do not match.
• Assuming AB = BA without checking.
• Concluding AB = O implies A = O or B = O.
• Reversing the order when multiplying by an inverse matrix.
• Forgetting that k(AB) = (kA)B = A(kB) is valid, but (kA)(lB) = klAB.

• 将不同阶数的矩阵相加。加法要求阶数相同。
• 将内维度不匹配的矩阵相乘。
• 不检查就假设 AB = BA。
• 由 AB = O 断言 A = O 或 B = O。
• 在乘以逆矩阵时颠倒顺序。
• 忘记 k(AB) = (kA)B = A(kB) 成立，且 (kA)(lB) = klAB。

Always check the order of matrices before carrying out a calculation, and preserve the order of multiplication when using inverse matrices.

在进行计算之前始终检查矩阵的阶数，并在使用逆矩阵时保持乘法的顺序不变。

---

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）