Matrix transformations, inverses and eigenvalues · Transformaciones matriciales, inversas y autovalores
| English | Español |
|---|---|
| determinant/dɪˈtɜːmɪnənt/ | determinante |
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? · ¿Cuál descripción define correctamente un determinante?
A scalar that determines whether a square matrix is invertible and, in two dimensions, its signed area factor. · Un escalar que determina si una matriz cuadrada es invertible y, en dos dimensiones, su factor de área con signo.
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 · Transformaciones matriciales, inversas y autovalores
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 · Una matriz 2×2 [[a,b],[c,d]] tiene determinante ad-bc e inversa [[d,-b],[-c,a]]/(ad-bc) si el determinante es distinto de cero
Compare the model with the worked case and explain one change. · Compara el modelo con el caso resuelto y explica un cambio.
Find the determinant of [[2,1],[0,3]]. · Encuentre el determinante de [[2,1],[0,3]].
The determinant is 2×3-1×0=6. · El determinante es 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). · Encuentre el componente x después de que esa matriz transforma (1,2).
The first transformed coordinate is 2×1+1×2=4. · La primera coordenada transformada es 2×1+1×2=4.
Every square matrix has an inverse. · Toda matriz cuadrada tiene una inversa.
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 multiplicación de matrices usualmente no es conmutativa. Un determinante de cero significa que la transformación pierde una dimensión y no tiene inversa. Los autovalores y la diagonalización son temas avanzados, no contenido ordinario de vectores en 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. · Encuentra el autovalor mayor de esa matriz.
This triangular matrix has eigenvalues given by its diagonal entries 2 and 3, so the larger is 3. · Esta matriz triangular tiene autovalores dados por sus entradas diagonales 2 y 3, por lo que el mayor es 3.
Match each part of a complete solution to its purpose. · Emparejar cada parte de una solución completa con su propósito.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Una suposición justifica el modelo; una comprobación verifica el resultado; la interpretación lo conecta con la pregunta.
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.