The Cayley-Hamilton Theorem states that every square matrix satisfies its own characteristic equation. If the characteristic polynomial of a matrix A is p(λ), then substituting the matrix into the polynomial gives p(A) = 0.
It allows for: Simplification of higher matrix powers. Computing inverses of matrices. Solving systems of differential equations. Reducing computational complexity in engineering and physics applications.
No. The theorem only applies to square matrices because only square matrices have characteristic polynomials (defined using determinants).
Yes, the theorem is valid for matrices over any field, including real numbers and complex numbers.
Absolutely. Once you derive the characteristic polynomial, you can express higher powers of A in terms of lower ones using the theorem—significantly simplifying calculations.
Yes. You can apply it to matrices with symbolic entries (like a, b, c) and use algebraic identities to verify or manipulate expressions.
The minimal polynomial of a matrix also annihilates the matrix (i.e., m(A) = 0) and divides the characteristic polynomial. The Cayley-Hamilton theorem guarantees that the characteristic polynomial does this as well, though it may not be minimal.
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Cayley-Hamilton Theorem
The Cayley-Hamilton Theorem is a key result in linear algebra stating that every square matrix satisfies its own characteristic equation. Discovered by Cayley and Hamilton, this theorem is both theoretically significant and practically useful. It simplifies complex matrix operations, helps compute matrix powers and inverses, and has applications in engineering, physics, and computer science.
1.0State the Cayley-Hamilton Theorem
Theorem:Every square matrix satisfies its own characteristic equation.
Let A be an n × n square matrix and let p(λ)=det(λI−A) be its characteristic polynomial. Then, p(A) = 0
That is, if the characteristic polynomial is:
p(λ)=λn+an−1λn−1+⋯+a1λ+a0
then,
An+an−1An−1+⋯+a1A+a0I=0
2.0Cayley-Hamilton Theorem Proof
Although there are several proofs, here’s a sketch using linear algebra and the concept of adjugate:
Start with the identity:
(λI−A)adj(λI−A)=det(λI−A)I
Replace λ with A (though not algebraically valid, it’s made rigorous using polynomial substitution).
Because adj(λI − A) is a matrix of polynomials in λ, substituting A into the polynomial gives p(A) = 0.
This proof hinges on the fundamental identity involving the determinant and adjugate of a matrix.
3.0Cayley-Hamilton Theorem for 3 x 3 Matrix
Let
A=21−1−11−1102
Step 1: Find the characteristic polynomial
det(λI−A)=λ−2−111λ−11−10λ=λ3−3λ2+2λ+4
Step 2: Apply the Cayley-Hamilton theorem
A3−3A2+2A+4I=0
This equation is valid for matrix A, and can be used to find other properties of the matrix.
4.0Application of Cayley-Hamilton Theorem
The theorem is more than a theoretical curiosity—it has practical applications:
Matrix Inversion: Especially when Gaussian elimination is cumbersome.
Solving Differential Equations: When systems are represented in matrix form.
Power of Matrices: Computing Anwithout repeated multiplication.
Control Theory: Used in state space representation of systems.
5.0Cayley-Hamilton Theorem to Find Inverse
Given:
An+an−1An−1+⋯+a1A+a0I=0
Rewriting this:
a0I=−(An+an−1An−1+⋯+a1A)
Multiply both sides by A−1(if it exists), and isolate A−1. For a 2 x 2 or 3 x 3 matrix, this method provides a neat way to find the inverse without determinants.
6.0Solved Examples on Cayley-Hamilton Theorem
Example 1: Verify the Cayley-Hamilton theorem for the matrix
Example 4: Verify the Cayley-Hamilton theorem for the matrix
A=[3−112]
Solution:
Step 1: Find the Characteristic Polynomialdet(λI−A)=λ−31−1λ−2=(λ−3)(λ−2)+1=λ2−5λ+7So the characteristic equation is:=(λ−3)(λ−2)+1=λ2−5λ+7Step 2: Apply the TheoremA2−5A+7I=0Compute:A2=[8−553],5A=[15−5510],7I=[7007]A2−5A+7I=[8−553]−[15−5510]+[7007]=[0000]
Example 5: Use the Cayley-Hamilton theorem to find the inverse of the matrix
A=[4332]
Solution:
Step 1: Characteristic Polynomialλ2−6λ+1⇒A2−6A+I=0Rewriting:A2−6A=−I⇒A(A−6I)=−I⇒A−1=−(A−6I)A−6I=[−233−4]⇒A−1=[2−3−34]Inverse Found Using Cayley-Hamilton.
7.0Practice Questions on Cayley-Hamilton Theorem
State the Cayley-Hamilton theorem. What is its significance in linear algebra?
Prove that the matrix
A=[ 1 011 ]
satisfies its characteristic equation.
What is the characteristic polynomial of the matrix
A=[ 0−21−3 ]
and verify the Cayley-Hamilton theorem for it.
Can the Cayley-Hamilton theorem be applied to non-square matrices? Justify your answer.
Use the Cayley-Hamilton theorem to compute A3 for
A=[ 1 110 ]
Use the Cayley-Hamilton theorem to find the inverse of
A=[ 2−1−12 ](Hint:UseA2−4A+3I=0)
Given,
A=[ 4 213 ]
verify the Cayley-Hamilton theorem and use it to find A^5.
Prove that the 3×3 matrix
A= 0 0−6101101−6 satisfies its own characteristic equation.
A matrix A satisfies the equation
A2+A−2I=0
Use this to find A−1.
Let A= 3 2 2122−1−10
Find the characteristic polynomial of A, verify the Cayley-Hamilton theorem, and use it to compute A−1.