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.
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.
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.
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.
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.
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 pure 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.