📚 Matrix Arithmetic | 矩阵运算
Matrix arithmetic is a fundamental topic in AQA A-Level Mathematics and Further Mathematics. It includes addition, subtraction, scalar multiplication and matrix multiplication, together with the key properties that govern these operations. A secure understanding of matrix arithmetic is essential for solving systems of equations, transformations and many applied problems.
矩阵运算是 AQA A-Level 数学及进阶数学中的基础内容。它包括矩阵的加法、减法、标量乘法与矩阵乘法，以及支配这些运算的重要性质。扎实掌握矩阵运算对于求解方程组、变换问题和许多应用问题都至关重要。
1. Matrix Notation and Order | 矩阵记号与阶
A matrix is usually denoted by a capital letter such as A, B or M, and its entries are written as lower-case letters with a row index and a column index. The entry in row i and column j is written aᵢⱼ.
矩阵通常用大写字母如 A、B 或 M 表示，其元素用小写字母加上行指标和列指标表示。第 i 行第 j 列的元素记作 aᵢⱼ。
s and n columns. The order, or dimension, of a matrix is written as m × n, read as m by n.
m × n 矩阵有 m 行和 n 列。矩阵的阶数或维数写作 m × n，读作 m 乘 n。
A square matrix has the same number of rows and columns, and a row matrix has only one row, while a column matrix has only one column.
方阵的行数和列数相同，行矩阵只有一行，列矩阵只有一列。
A = [ a₁₁ a₁₂ ; a₂₁ a₂₂ ]
A 2 × 2 matrix with entries aᵢⱼ
2. Matrix Addition and Subtraction | 矩阵加减法
Addition and subtraction are performed entry by entry, and both matrices must have exactly the same order. If the orders differ, the operations are undefined.
加法和减法按元素逐个进行，且两个矩阵必须具有完全相同的阶数。如果阶数不同，则这两个运算没有定义。
If A = [aᵢⱼ] and B = [bᵢⱼ], then A + B = [aᵢⱼ + bᵢⱼ] and A − B = [aᵢⱼ − bᵢⱼ].
若 A = [aᵢⱼ] 且 B = [bᵢⱼ]，则 A + B = [aᵢⱼ + bᵢⱼ]，A − B = [aᵢⱼ − bᵢⱼ]。
e, if A = [2 1; 3 4] and B = [5 0; −1 2], then A + B = [7 1; 2 6] and A − B = [−3 1; 4 2].
例如，若 A = [2 1; 3 4]，B = [5 0; −1 2]，则 A + B = [7 1; 2 6]，A − B = [−3 1; 4 2]。
commutative, so A + B = B + A, and subtraction is simply addition of the negative matrix.
加法满足交换律，因此 A + B = B + A，而减法可以看作加上负矩阵的加法。
3. Scalar Multiplication | 标量乘法
A scalar is just a number. To multiply a matrix by a scalar k, multiply every entry of the matrix by k.
标量就是一个数。矩阵乘以标量 k 时，将矩阵的每一个元素都乘以 k。
If A = [aᵢⱼ], then kA = [kaᵢⱼ]. For example, 3A = 3[aᵢⱼ] = [3aᵢⱼ].
若 A = [aᵢⱼ]，则 kA = [kaᵢⱼ]。例如，3A = 3[aᵢⱼ] = [3aᵢⱼ]。
over matrix addition: k(A + B) = kA + kB, and (k + p)A = kA + pA for any scalars k and p.
标量乘法对矩阵加法满足分配律：k(A + B) = kA + kB，并且对任意标量 k 和 p 有 (k + p)A = kA + pA。
The negative of a matrix is obtained by multiplying by −1, so −A = [−aᵢⱼ].
矩阵的负矩阵可通过乘以 −1 得到，因此 −A = [−aᵢⱼ]。
4. Matrix Multiplication: Definition | 矩阵乘法定义
Matrix multiplication is not performed entry by entry. To multiply A by B, the number of columns of A must equal the number of rows of B.
矩阵乘法不是逐元素相乘。要用 A 乘以 B，A 的列数必须等于 B 的行数。
If A is m × n and B is n × p, then the product C = AB is m × p. Each entry cᵢⱼ ntries of row i of A by the corresponding entries of column j of B and adding the results.
若 A 为 m × n，B 为 n × p，则乘积 C = AB 为 m × p。每个元素 cᵢⱼ 通过将 A 的第 i 行元素与 B 的第 j 列对应元素相乘后求和得到。
cᵢⱼ = aᵢ₁b₁ⱼ + aᵢ₂b₂ⱼ + ⋯ + aᵢₙbₙⱼ
Entry cᵢⱼ is the dot product of row i of A and column j of B
The product AB is defined only when the number of columns of A equals the number of rows of B. The resulting matrix takes the number of rows from A and the number of columns from B.
乘积 AB 只有在 A 的列数等于 B 的行数时才有定义。所得矩阵的行数来自 A，列数来自 B。
5. Worked Example of Matrix Multiplication | 矩阵乘法例题
Let A = [1 2; 3 4] and B = [5 6; 7 8].
设 A = [1 2; 3 4]，B = [5 6; 7 8]。
s 2 × 2 and B is 2 × 2, the product AB is also 2 × 2. Each entry is calculated as follows.
由于 A 为 2 × 2，B 为 2 × 2，乘积 AB 也是 2 × 2。每个元素计算如下。
AB = [ 1×5 + 2×7 1×6 + 2×8 ; 3×5 + 4×7 3×6 + 4×8 ] = [ 19 22 ; 43 50 ]
Row-by-column calculation of AB
(2 × 7) = 19, and the entry in the second row and first column is (3 × 5) + (4 × 7) = 43.
第一行第一列的元素为 (1 × 5) + (2 × 7) = 19，第二行第一列的元素为 (3 × 5) + (4 × 7) = 43。
It is useful to write out the row-by-column products in full during an exam, rather than trying to do the multiplication mentally.
考试时建议把行乘列的乘积完整写出，而不要仅凭心算完成乘法。
6. Identity Matrix | 单位矩阵
The identity matrix, written I, acts like the number 1 in matrix multiplication. It is a square matrix with 1s on the main diagonal and 0s everywhere else.
单位矩阵记作 I，在矩阵乘法中类似于数字 1。它是方阵，主对角线上的元素为 1，其余元素均为 0。
y matrix A of appropriate order, AI = IA = A. The 2 × 2 identity matrix is I = [1 0; 0 1].
对于任何阶数匹配的矩阵 A，都有 AI = IA = A。2 × 2 单位矩阵为 I = [1 0; 0 1]。
The 3 × 3 identity matrix is I = [1 0 0; 0 1 0; 0 0 1].
3 × 3 单位矩阵为 I = [1 0 0; 0 1 0; 0 0 1]。
Multiplying by I on the left or the right leaves the original matrix unchanged, provided the orders are compatible.
只要阶数兼容，左乘或右乘单位矩阵都不会改变原矩阵。
7. Zero Matrix and Arithmetic Properties | 零矩阵与运算性质
It is the additive identity for matrices, so A + 0 = A for any matrix A of the same order.
零矩阵记作 0，所有元素都为 0。它是矩阵的加法单位元，因此对于任何阶数相同的矩阵 A，都有 A + 0 = A。
Every matrix A has an additive inverse, namely −A, and A + (−A) = 0.
每个矩阵 A 都有一个加法逆元，即 −A，且 A + (−A) = 0。
0 or B = 0. In matrix algebra, two non-zero matrices can multiply to give the zero matrix.
然而，AB = 0 并不一定推出 A = 0 或 B = 0。在矩阵代数中，两个非零矩阵相乘可以得到零矩阵。
This is an important difference from ordinary number arithmetic, and it means that we cannot always cancel common matrix factors.
这是矩阵运算与普通数字运算的一个重要区别，也意味着我们不能总是约去共同的矩阵因子。
8. Non-Commutativity of Matrix Multiplication | 矩阵乘法不可交换
In general, matrix multiplication is not commutative, so AB ≠ BA. Even when both products exist, they may have different orders or different entries.
一般来说，矩阵乘法不满足交换律，因此 AB ≠ BA。即使两个乘积都存在，它们的阶数或元素也可能不同。
For example, let A = [0 1; 0 0] and B = [0 0; 1 0].
例如，设 A = [0 1; 0 0]，B = [0 0; 1 0]。
AB = [ 1 0 ; 0 0 ]
BA = [ 0 0 ; 0 1 ]
Here both AB and BA exist and are both 2 × 2, but AB ≠ BA.
这里 AB 和 BA 都存在且都是 2 × 2，但 AB ≠ BA。
There are special cases where two matrices commute, for example when one of them is the identity matrix or when one is a scalar multiple of the identity matrix. However, the general rule is that order matters in matrix multiplication.
有些特殊情况下两个矩阵可以交换，例如其中一个为单位矩阵，或其中一个为单位矩阵的标量倍。但一般规律是，矩阵乘法中顺序十分重要。
9. Associative and Distributive Laws | 结合律与分配律
ative: (AB)C = A(BC), provided the orders of the matrices allow the products to be formed.
矩阵乘法满足结合律：(AB)C = A
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