Characteristic Polynomial
Eigenvalues are roots of det(A − λI).
Key idea
Eigenvalues are roots of det(A − λI).
In depth
Eigenvalues solve det(A − λI) = 0. For a 2 × 2 matrix it is λ² − (trace)λ + det = 0. The roots can be complex.
Worked example
A = [4 1; 2 3]: λ² − 7λ + 10 = 0, so λ = 2, 5.
Formula
Real-world uses
- Finding growth rates of populations and stability of systems in ecology and control
Common mistake
Computing det(A + λI) or sign errors with the trace.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Characteristic Polynomial?
Eigenvalues are roots of det(A − λI). It is a topic at the expert level in Linear Algebra.
What is the formula for Characteristic Polynomial?
The key formula is det(A − λI) = 0.
Where is Characteristic Polynomial used in real life?
Real-world examples: Finding growth rates of populations and stability of systems in ecology and control.
What should I learn before Characteristic Polynomial?
Make sure you are comfortable with Eigenvalues and Eigenvectors, the previous topic in Linear Algebra.
What is a common mistake with Characteristic Polynomial?
Computing det(A + λI) or sign errors with the trace.
How can I practise Characteristic Polynomial?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Characteristic Polynomial?
Move on to Eigenspaces next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.