Matrix Arithmetic | 矩阵运算

📚 Matrix Arithmetic | 矩阵运算

Matrices are among the most powerful tools in A-Level mathematics. They allow us to organise data compactly, perform geometric transformations, and solve systems of linear equations with efficiency and elegance. This article covers all the matrix arithmetic skills required by the AQA A-Level Mathematics specification, with step-by-step worked examples throughout.

矩阵是A-Level数学中最强大的工具之一。它使我们能够以紧凑的方式整理数据、执行几何变换，并高效而优雅地求解线性方程组。本文将全面讲解AQA A-Level数学大纲所要求的矩阵运算技能，并配有逐步演算的例题。

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1. What Is a Matrix? | 什么是矩阵？

A matrix is a rectangular array of numbers arranged in rows and columns. For example:

矩阵是按行和列排列成矩形阵列的一组数字。例如：

A = [ 2 -1 ]
[ 3 4 ]

The order (or dimension) of a matrix is stated as rows × columns. The matrix A above has order 2×2, read as “two by two”. A matrix with m rows and n columns is said to have order m×n.

矩阵的阶数（或维度）以行×列来表示。上面的矩阵A是2×2阶，读作”二乘二”。有m行n列的矩阵称为m×n阶矩阵。

Each entry is referenced using subscripts: aᵢⱼ denotes the element in the i-th row and j-th column. In matrix A above, a₁₁ = 2, a₁₂ = -1, a₂₁ = 3 and a₂₂ = 4.

每个元素用下标引用：aᵢⱼ 表示第i行第j列的元素。在上面的矩阵A中，a₁₁ = 2，a₁₂ = -1，a₂₁ = 3，a₂₂ = 4。

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2. Matrix Addition and Subtraction | 矩阵的加法与减法

Two matrices can be added or subtracted only if they have the same order. The result is obtained by adding or subtracting the corresponding entries.

两个矩阵只有在阶数相同时才能进行加法或减法运算。结果是逐项相加或相减得到的矩阵。

For example, let A = [ 1 2 ; 3 4 ] and B = [ 5 6 ; 7 8 ]:

例如，设 A = [ 1 2 ; 3 4 ]，B = [ 5 6 ; 7 8 ]：

A + B = [ 1+5 2+6 ; 3+7 4+8 ] = [ 6 8 ; 10 12 ]

A – B = [ 1-5 2-6 ; 3-7 4-8 ] = [ -4 -4 ; -4 -4 ]

Note that matrix addition is commutative (A + B = B + A) and associative, just like ordinary number addition. However, subtraction is not commutative.

注意，矩阵加法满足交换律（A + B = B + A）和结合律，这与普通数的加法一样。但减法不满足交换律。

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3. Scalar Multiplication | 标量乘法

A matrix can be multiplied by a scalar (a single number). Each entry in the matrix is multiplied by that scalar.

矩阵可以与一个标量（单个数值）相乘。矩阵中的每一个元素都要乘以这个标量。

For example, if k = 2 and A = [ 1 -2 ; 3 0 ], then:

例如，若 k = 2，A = [ 1 -2 ; 3 0 ]，则：

2A = [ 2×1 2×(-2) ; 2×3 2×0 ] = [ 2 -4 ; 6 0 ]

Scalar multiplication is distributive over matrix addition: k(A + B) = kA + kB.

标量乘法对矩阵加法满足分配律：k(A + B) = kA + kB。

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4. Matrix Multiplication | 矩阵乘法

Matrix multiplication is more subtle. For the product AB to be defined, the number of columns of A must equal the number of rows of B. If A is m×n and B is n×p, then AB is m×p.

矩阵乘法更为微妙。乘积AB有定义的前提是：A的列数必须等于B的行数。若A是m×n阶，B是n×p阶，则AB是m×p阶。

The element in row i and column j of AB is found by taking the dot product of row i of A with column j of B.

AB中第i行第j列的元素，是将A的第i行与B的第j列作点积运算得到。

(AB)ᵢⱼ = Σₖ aᵢₖ bₖⱼ

Worked example: let A = [ 2 1 ; 3 4 ] and B = [ 1 2 ; 0 3 ]. Both are 2×2, so AB is 2×2.

例题：设 A = [ 2 1 ; 3 4 ]，B = [ 1 2 ; 0 3 ]。两者都是2×2阶，所以AB也是2×2阶。

AB = [ 2×1+1×0 2×2+1×3 ; 3×1+4×0 3×2+4×3 ]

AB = [ 2 7 ; 3 18 ]

Now compare this with BA:

现在将它与BA

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