Matrix transformations, inverses and eigenvalues · Transformations matricielles, inverses et valeurs propres
| English | Français |
|---|---|
| determinant/dɪˈtɜːmɪnənt/ | déterminant |
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? · Quelle description définit correctement le déterminant ?
A scalar that determines whether a square matrix is invertible and, in two dimensions, its signed area factor. · Un scalaire qui détermine si une matrice carrée est inversible et, en deux dimensions, son facteur d'aire signée.
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 · Transformations matricielles, inverses et valeurs propres
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 · Une matrice 2×2 [[a,b],[c,d]] a pour déterminant ad-bc et pour inverse [[d,-b],[-c,a]]/(ad-bc) si le déterminant est non nul.
Compare the model with the worked case and explain one change. · Compare le modèle avec l'exemple résolu et explique un changement.
Find the determinant of [[2,1],[0,3]]. · Trouvez le déterminant de [[2,1],[0,3]].
The determinant is 2×3-1×0=6. · Le déterminant est 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). · Trouvez la composante x après que cette matrice transforme (1,2).
The first transformed coordinate is 2×1+1×2=4. · La première coordonnée transformée est 2×1+1×2=4.
Every square matrix has an inverse. · Toute matrice carrée possède un 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. · La multiplication matricielle n'est généralement pas commutative. Un déterminant nul signifie que la transformation perd une dimension et n'a pas d'inverse. Les valeurs propres et la diagonalisation relèvent d'un niveau avancé, hors du programme ordinaire de vecteurs au GCSE.
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. · Trouver la plus grande valeur propre de cette matrice.
This triangular matrix has eigenvalues given by its diagonal entries 2 and 3, so the larger is 3. · Cette matrice triangulaire a pour valeurs propres ses éléments diagonaux 2 et 3 ; la plus grande est donc 3.
Match each part of a complete solution to its purpose. · Associer chaque partie d'une solution complète à son but.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Une hypothèse justifie le modèle ; une vérification teste le résultat ; l'interprétation le relie à la question.
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.