Matrix transformations, inverses and eigenvalues
Diperkenalkan| English | Makna |
|---|---|
| determinant/dɪˈtɜːmɪnənt/ | determinan |
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.
The method
- 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.
Why does a zero determinant mean a 2×2 matrix has no inverse?
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The determinant is the signed area scale factor. det=0 collapses a dimension - distinct points share an image - so no inverse transformation can separate them, and division by ad-bc fails.
A worked case
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]].
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The determinant is 2×3-1×0=6.
Where it goes wrong
- 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.
Not true: Every square matrix has an inverse. A square matrix with determinant 0 is singular and has no inverse.
Find the x-component after that matrix transforms (1,2).
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The first transformed coordinate is 2×1+1×2=4.
Every square matrix has an inverse.
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A square matrix with determinant 0 is singular and has no inverse.
Applying it
- 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.
Find the larger eigenvalue of that matrix.
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This triangular matrix has eigenvalues given by its diagonal entries 2 and 3, so the larger is 3.
Match each matrix concept to its role.
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Matrix products act right to left, and an inverse exists exactly when the determinant is nonzero.
In your course
- Pearson Edexcel International A Level Pure Mathematics, unit FP3.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Determinant: A scalar that determines whether a square matrix is invertible and, in two dimensions, its signed area factor.
- 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.
- Watch for: matrix multiplication is usually not commutative.
Write from a determinant of 10
- The matrix has rows (5, 1) and (0, 2).
- Find the determinant.
What is the determinant of the matrix with rows (5, 1) and (0, 2)?
5 times 2 is 10.
What is the determinant?
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The diagonal product is 10. Other options — 6 (the lesson determinant); 5 (the larger eigenvalue).