Matrix transformations, inverses and eigenvalues · 矩阵变换、逆矩阵及特征值
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| determinant/dɪˈtɜːmɪnənt/ | 行列式 | háng liè shì |
Where do the basis vectors go?
- A map stretches one direction and shears another. A matrix records the transformation of each basis vector.
- This lesson studies determinant 行列式: A scalar that determines whether a square matrix is invertible and, in two dimensions, its signed area factor.
基向量去了哪里？
- 映射会拉伸一个方向并剪切另一个方向。矩阵记录了每个基向量的变换过程。
- 本课学习 行列式 determinant：一个标量，用于判定方阵是否可逆，在二维空间中代表有符号面积因子。
Choose the mathematical structure
- A 2×2 matrix [[a,b],[c,d]] has determinant ad-bc and inverse [[d,-b],[-c,a]]/(ad-bc) if the determinant is nonzero. Transform column vectors by left multiplication. The rightmost matrix acts first in a product.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
选择数学结构
- 一个 2×2 矩阵 [[a,b],[c,d]] 的行列式为 ad-bc，若行列式非零，则其逆矩阵为 [[d,-b],[-c,a]]/(ad-bc)。通过左乘变换列向量。在乘积中，最右侧的矩阵先作用。
- 计算前请先明确允许的输入项和单位。方程应表达关系本身，而不仅仅是记录计算器按键过程。
Which description correctly defines determinant? · 下列哪项描述正确定义了“行列式”？
A scalar that determines whether a square matrix is invertible and, in two dimensions, its signed area factor. · 一个标量，用于判定方阵是否可逆，并在二维中代表其带符号的面积缩放因子。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For A=[[2,1],[0,3]], det A=6. A(1,2)=(4,6). Its inverse is [[1/2,-1/6],[0,1/3]]. The characteristic equation is (2-λ)(3-λ)=0, so the eigenvalues are 2 and 3.
通过验证案例进行推导
- 将结果与初始量进行比对。代入原始关系式，或在适当情况下对比图表与数值答案。
对于 A=[[2,1],[0,3]]，行列式 det A=6。A(1,2)=(4,6)。其逆矩阵为 [[1/2,-1/6],[0,1/3]]。特征方程为 (2-λ)(3-λ)=0，故特征值为 2 和 3。
Matrix transformations, inverses and eigenvalues · 矩阵变换、逆矩阵及特征值
A 2×2 matrix [[a,b],[c,d]] has determinant ad-bc and inverse [[d,-b],[-c,a]]/(ad-bc) if the determinant is nonzero · 一个 2×2 矩阵 [[a,b],[c,d]] 的行列式为 ad-bc；若行列式非零，其逆矩阵为 [[d,-b],[-c,a]]/(ad-bc)。
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
Find the determinant of [[2,1],[0,3]]. · 求 [[2,1],[0,3]] 的行列式。
The determinant is 2×3-1×0=6. · 该行列式的值为 2×3-1×0=6。
Test a tempting shortcut
- Matrix multiplication is usually not commutative. A determinant of zero means the transformation loses a dimension and has no inverse. Eigenvalues and diagonalisation are advanced scope, not ordinary GCSE vector content.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every square matrix has an inverse. This claim is false. Explain which definition or assumption it violates.
检验一个诱人的捷径
- 矩阵乘法通常不满足交换律。行列式为零意味着变换丢失了一个维度且不可逆。特征值与对角化属于高阶范畴，不属于常规 GCSE 向量内容。
- 当捷径失效时，找出其违背的假设。保留精确值直到题目要求的最终舍入步骤。
每一个方阵都有逆矩阵。此说法错误。请解释它违反了哪一定义或假设。
Find the x-component after that matrix transforms (1,2). · 求该矩阵变换后向量的 x 分量（1,2）。
The first transformed coordinate is 2×1+1×2=4. · 第一个变换后的坐标为 2×1+1×2=4。
Every square matrix has an inverse. · 每个方阵都有逆矩阵。
Matrix multiplication is usually not commutative. A determinant of zero means the transformation loses a dimension and has no inverse. Eigenvalues and diagonalisation are advanced scope, not ordinary GCSE vector content. · 矩阵乘法通常不满足交换律。行列式为零意味着该变换丢失了一个维度，因此不存在逆矩阵。特征值与对角化属于进阶内容，不属于常规 GCSE 向量范畴。
Interpret a new situation
- Find an eigenvector by solving (A-λI)v=0 with v≠0. Explain the geometrical meaning: this vector keeps its line direction under the transformation.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
解读新情境
- 通过求解 (A-λI)v=0（其中 v≠0）来找到特征向量。解释其几何意义：该向量在变换下保持其所在直线的方向不变。
- 完整解答需给出数学结果并解释其含义。检查其在所述背景下是否可行。
Find the larger eigenvalue of that matrix. · 求该矩阵的较大特征值。
This triangular matrix has eigenvalues given by its diagonal entries 2 and 3, so the larger is 3. · 该三角矩阵的特征值由其对角线元素 ⟨2⟩ 和 ⟨3⟩ 给出，因此较大的值为 ⟨3⟩。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- edexcel IAL further mathematics; official unit FP3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A scalar that determines whether a square matrix is invertible and, in two dimensions, its signed area factor. Choose the relationship, show the method, check its assumptions and interpret the result.
将其应用于你的课程
- Edexcel IAL 进阶数学；官方单元 FP3。其他单元的补充内容在范围审查中已列出；不属于额外的加分要求。
- 在给出最终答案前先展示解题方法，并遵循试卷的计算器及公式使用规则。通过定位第一个无效步骤来复盘错误答案。
一个标量，用于判定方阵是否可逆，在二维空间中代表有符号面积因子。请选择关系式，展示方法，验证其假设并解读结果。