Cayley–Hamilton Theorem
Every matrix satisfies its characteristic equation.
Key idea
Every matrix satisfies its characteristic equation.
In depth
Every square matrix satisfies its own characteristic polynomial: p(A) = 0. It gives a way to compute inverses and high powers by reducing them to lower powers.
Worked example
For A = [4 1; 2 3]: A² − 7A + 10I = 0, so A⁻¹ = (7I − A)/10.
Formula
Real-world uses
- Computing matrix powers quickly in control engineering and economics models
Common mistake
Substituting a scalar for the constant term instead of the identity matrix.
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 Cayley–Hamilton Theorem?
Every matrix satisfies its characteristic equation. It is a topic at the expert level in Linear Algebra.
What is the formula for Cayley–Hamilton Theorem?
The key formula is p(A) = 0.
Where is Cayley–Hamilton Theorem used in real life?
Real-world examples: Computing matrix powers quickly in control engineering and economics models.
What should I learn before Cayley–Hamilton Theorem?
Make sure you are comfortable with Trace, the previous topic in Linear Algebra.
What is a common mistake with Cayley–Hamilton Theorem?
Substituting a scalar for the constant term instead of the identity matrix.
How can I practise Cayley–Hamilton Theorem?
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 Cayley–Hamilton Theorem?
Move on to Jordan Canonical Form 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.