📚 Matrix Algebra for AQA A-Level Mathematics | AQA A-Level 数学：矩阵代数
Matrix algebra is a compact and powerful tool for handling systems of linear equations, geometric transformations, and many problems in pure and applied mathematics. In AQA A-Level Mathematics, you need to be confident with matrix operations, determinants, inverses, and their geometric meaning.
矩阵代数是处理线性方程组、几何变换以及纯数学和应用数学中许多问题的一种简洁而有力的工具。在 AQA A-Level 数学中，你需要熟练掌握矩阵运算、行列式、逆矩阵及其几何意义。
1. What is a matrix? | 什么是矩阵？
and n is the number of columns. For example, a 2 × 2 matrix has two rows and two columns.
矩阵是由数字（称为元素）按行和列排列成的矩形阵列。矩阵的阶数写作 m × n，其中 m 是行数，n 是列数。例如，一个 2 × 2 矩阵有两行两列。
Matrix entries are usually denoted by a subscript: aᵢⱼ is the element in row i and column j. Two matrices are equal only when they have the same order and all corresponding entries are equal.
矩阵元素通常用下标表示：aᵢⱼ 表示第 i 行第 j 列的元素。两个矩阵相等当且仅当它们具有相同的阶数并且所有对应元素都相等。
2. Matrix addition and subtraction | 矩阵加减法
Matrices can be added or subtracted only when they have the same order. You add or subtract corresponding entries one by one. The result has the same order as the original matrices.
矩阵只有在阶数相同时才能相加或相减。你只需逐一对应元素进行加减。结果矩阵的阶数与原矩阵相同。
tive, so for conformable matrices A, B and C: A + B = B + A and (A + B) + C = A + (B + C).
加法满足交换律和结合律，因此对于可加矩阵 A、B 和 C：A + B = B + A，(A + B) + C = A + (B + C)。
3. Scalar multiplication | 标量乘法
To multiply a matrix by a scalar k, multiply every entry of the matrix by k. If k is negative, every entry changes sign; if k is zero, the result is the zero matrix of the same order.
要将矩阵乘以标量 k，只需将矩阵的每个元素都乘以 k。如果 k 是负数，每个元素变号；如果 k 是零，结果就是同阶的零矩阵。
) = kA + kB. This operation is useful when combining transformations or scaling data sets.
标量乘法对矩阵加法满足分配律：k(A + B) = kA + kB。该运算在组合变换或缩放数据时非常有用。
4. Matrix multiplication | 矩阵乘法
lumns of A equals the number of rows of B. If A is m × n and B is n × p, then AB is m × p.
矩阵乘法不是逐元素相乘。乘积 AB 只有当 A 的列数等于 B 的行数时才有定义。如果 A 是 m × n 矩阵，B 是 n × p 矩阵，那么 AB 是 m × p 矩阵。
of row i of A by the corresponding entries of column j of B and adding the products: (AB)ᵢⱼ = ∑ₖ aᵢₖ bₖⱼ.
AB 中第 i 行第 j 列的元素，是将 A 的第 i 行各元素与 B 的第 j 列对应元素相乘后求和得到：(AB)ᵢⱼ = ∑ₖ aᵢₖ bₖⱼ。
r, it is not generally commutative: AB is usually different from BA. Always check the order.
矩阵乘法满足结合律 A(BC) = (AB)C，并对加法满足分配律 A(B + C) = AB + AC。但矩阵乘法通常不满足交换律：AB 一般不等于 BA。务必检查相乘顺序。
5. The identity matrix | 单位矩阵
ke the number 1 in matrix multiplication: AI = IA = A for any conformable square matrix A.
单位矩阵 I 是一个方阵，主对角线上的元素为 1，其余元素为 0。对于 2 × 2 矩阵，I = (1 0; 0 1)。它在矩阵乘法中的作用类似于数字 1：对任意可乘的方阵 A，有 AI = IA = A。
The identity matrix is crucial for defining inverse matrices and for understanding transformations that leave vectors unchanged.
单位矩阵对于定义逆矩阵和理解使向量保持不变的变换至关重要。
6. Determinants of 2 × 2 and 3 × 3 matrices | 二阶与三阶行列式
The determinant is a scalar value that can be calculated from a square matrix. It determines whether a matrix is invertible and gives the area or volume scale factor of the associated linear transformation.
行列式是一个可以从方阵计算出的标量值。它决定矩阵是否可逆，并给出相关线性变换的面积或体积缩放因子。
For a 2 × 2 matrix A = (a b; c d), the determinant is det(A) = ad - bc.
对于 2 × 2 矩阵 A = (a b; c d)，行列式为 det(A) = ad - bc。
det(A) = |a b; c d| = ad - bc
matrix, expand along the first row using 2 × 2 minors. If A = (a b c; d e f; g h i), then:
对于 3 × 3 矩阵，沿第一行按 2 × 2 余子式展开。若 A = (a b c; d e f; g h i)，则：
det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
is singular and has no inverse. If det(A) ≠ 0, the matrix is non-singular and invertible.
如果 det(A) = 0，矩阵是奇异矩阵，不存在逆矩阵。如果 det(A) ≠ 0，矩阵是非奇异矩阵，可逆。
7. Inverse of a 2 × 2 matrix | 二阶矩阵的逆
es AA⁻¹ = A⁻¹A = I. Not every matrix has an inverse; only non-singular square matrices do.
方阵 A 的逆矩阵记作 A⁻¹，满足 AA⁻¹ = A⁻¹A = I。并非每个矩阵都有逆矩阵，只有非奇异方阵才有。
For a 2 × 2 matrix A = (a b; c d), the inverse is given by:
对于 2 × 2 矩阵 A = (a b; c d)，其逆矩阵为：
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图