Matrix transformations, inverses and eigenvalues
| English | 中文 | Pinyin |
|---|---|---|
| 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?
A scalar that determines whether a square matrix is invertible and, in two dimensions, its signed area factor.
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
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
Compare the model with the worked case and explain one change.
Find the determinant of [[2,1],[0,3]].
The determinant is 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).
The first transformed coordinate is 2×1+1×2=4.
Every square matrix has an inverse.
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.
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.
This triangular matrix has eigenvalues given by its diagonal entries 2 and 3, so the larger is 3.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- Current first-assessment-2021 Applications and Interpretation HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.