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 $p(\lambda )=\det (\lambda I-A)$ be its characteristic polynomial. Then, p(A) = 0 

That is, if the characteristic polynomial is:

$p(\lambda )={\lambda }^n+a_{n-1}{\lambda }^{n-1}+\cdots +a_1\lambda +a_0$

then,

$A^n+a_{n-1}{A}^{n-1}+\cdots +a_{1}A+a_{0}I=0$

2.0Cayley-Hamilton Theorem Proof 

Although there are several proofs, here’s a sketch using linear algebra and the concept of adjugate:

1. Start with the identity:

$(\lambda I-A)\mathrm{adj}(\lambda I-A)=\det (\lambda I-A)I$

2. Replace λ with A (though not algebraically valid, it’s made rigorous using polynomial substitution).
3. 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  

$A=\left[\begin{array}{ccc}2& -1& 1\\ 1& 1& 0\\ -1& -1& 2\end{array}\right]$

Step 1: Find the characteristic polynomial

$\det (\lambda I-A)=\left|\begin{array}{ccc}\lambda -2& 1& -1\\ -1& \lambda -1& 0\\ 1& 1& \lambda \end{array}\right|={\lambda }^3-3{\lambda }^2+2\lambda +4$

Step 2: Apply the Cayley-Hamilton theorem

$A^3-3A^2+2A+4I=0$

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 $A^n$without repeated multiplication.
• Control Theory: Used in state space representation of systems.

5.0Cayley-Hamilton Theorem to Find Inverse

Given:

$A^n+a_{n-1}{A}^{n-1}+\cdots +a_{1}A+a_{0}I=0$

Rewriting this: 

$a_{0}I=-\left(A^n+a_{n-1}{A}^{n-1}+\cdots +a_{1}A\right)$

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

$A=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]$

Solution: 

$$\begin{gathered} \text{Characteristic Polynomial:} \\ {\lambda }^2-4\lambda +3 \\ \text{Verification:} \\ {A}^2=\left[\begin{array}{cc}5& 4\\ 4& 5\end{array}\right],4A=\left[\begin{array}{cc}8& 4\\ 4& 8\end{array}\right],3I=\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right] \\ {A}^2-4A+3I=0 \end{gathered}$$

Example 2: Use the Cayley-Hamilton theorem to find the inverse of the matrix

Let  $A=\left[\begin{array}{cc}4& 3\\ 3& 2\end{array}\right]$

Solution: 

$$\begin{gathered} \text{Characteristic Polynomial:}{\lambda }^2-6\lambda +1\Rightarrow A^2-6A+I=0 \\ \text{Rewriting:} \\ {A}^2-6A=-I\Rightarrow A(A-6I)=-I\Rightarrow A^{-1}=-(A-6I) \\ {A}^{-1}=\left[\begin{array}{cc}2& -3\\ -3& 4\end{array}\right] \end{gathered}$$

Example 3: Verify the Cayley-Hamilton theorem for the matrix

Let  $A=\left[\begin{array}{ccc}1& 2& 0\\ 0& 1& 2\\ 0& 0& 1\end{array}\right]$

Solution: 

$$\begin{gathered} \text{Characteristic Polynomial:} \\ (\lambda -1{)}^3={\lambda }^3-3{\lambda }^2+3\lambda -1 \\ \text{Verification:} \\ \text{Calculate}{A}^2,A^3,\text{and verify:} \\ {A}^3-3A^2+3A-I=0 \end{gathered}$$

Example 4: Verify the Cayley-Hamilton theorem for the matrix  

$A=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$

Solution:

$$\begin{gathered} \text{Step 1: Find the Characteristic Polynomial}\det (\lambda I-A)=\left|\begin{array}{cc}\lambda -3& -1\\ 1& \lambda -2\end{array}\right|=(\lambda -3)(\lambda -2)+1={\lambda }^2-5\lambda +7 \\ \text{So the characteristic equation is:} \\ =(\lambda -3)(\lambda -2)+1={\lambda }^2-5\lambda +7 \\ \text{Step 2: Apply the Theorem} \\ {A}^2-5A+7I=0 \\ \text{Compute:} \\ {A}^2=\left[\begin{array}{cc}8& 5\\ -5& 3\end{array}\right],5A=\left[\begin{array}{cc}15& 5\\ -5& 10\end{array}\right],7I=\left[\begin{array}{cc}7& 0\\ 0& 7\end{array}\right] \\ {A}^2-5A+7I=\left[\begin{array}{cc}8& 5\\ -5& 3\end{array}\right]-\left[\begin{array}{cc}15& 5\\ -5& 10\end{array}\right]+\left[\begin{array}{cc}7& 0\\ 0& 7\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \end{gathered}$$

Example 5: Use the Cayley-Hamilton theorem to find the inverse of the matrix

$A=\left[\begin{array}{cc}4& 3\\ 3& 2\end{array}\right]$

Solution: 

$$\begin{gathered} \text{Step 1: Characteristic Polynomial} \\ {\lambda }^2-6\lambda +1\Rightarrow A^2-6A+I=0 \\ \text{Rewriting:} \\ {A}^2-6A=-I\Rightarrow A(A-6I)=-I\Rightarrow A^{-1}=-(A-6I) \\ A-6I=\left[\begin{array}{cc}-2& 3\\ 3& -4\end{array}\right]\Rightarrow A^{-1}=\left[\begin{array}{cc}2& -3\\ -3& 4\end{array}\right] \\ \text{Inverse Found Using Cayley-Hamilton.} \end{gathered}$$

7.0Practice Questions on Cayley-Hamilton Theorem

1. State the Cayley-Hamilton theorem. What is its significance in linear algebra?
2. Prove that the matrix

$A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$

satisfies its characteristic equation.

3. What is the characteristic polynomial of the matrix

$A=\left[\begin{array}{cc}0& 1\\ -2& -3\end{array}\right]$

and verify the Cayley-Hamilton theorem for it.

4. Can the Cayley-Hamilton theorem be applied to non-square matrices? Justify your answer.
5. Use the Cayley-Hamilton theorem to compute $A^3$ for

$A=\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$ 

6. Use the Cayley-Hamilton theorem to find the inverse of

$$\begin{gathered} A=\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right] \\ (Hint:Use{A}^2-4A+3I=0) \end{gathered}$$

7. Given,

$A=\left[\begin{array}{cc}4& 1\\ 2& 3\end{array}\right]$

verify the Cayley-Hamilton theorem and use it to find A^5.

8. Prove that the 3×3 matrix

$$\begin{gathered} A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -6& 11& -6\end{array}\right] \\ \text{satisfies its own characteristic equation.} \end{gathered}$$

9. A matrix A satisfies the equation

$A^2+A-2I=0$

Use this to find $A^{-1}$.

10. Let $A=\left[\begin{array}{ccc}3& 1& -1\\ 2& 2& -1\\ 2& 2& 0\end{array}\right]$

 Find the characteristic polynomial of A, verify the Cayley-Hamilton theorem, and use it to compute $A^{-1}$.