Cayley Hamilton Theorem: Statement, Theorem, Proof & Sample Questions

Cayley Hamilton Theorem states that all complex and real square matrices would satisfy their own characteristic polynomial equation. Cayley Hamilton Theorem is used in advanced linear algebra to simplify linear transformations. The distinctive polynomial of A is articulated as

 p(x) = det(xI_n – A)

if A is provided as n×n matrix, and I_n is the n×n identity matrix. Where the determinant operation is denoted by ‘det’ and for the scalar element of the base ring, the variable is taken as x. The determinant is also an n-th order monic polynomial in x as the entries of the matrix are linear or constant polynomials in x. Cayley Hamilton Theorem was given by two mathematicians Arthur Cayley and William Rowan Hamilton in the year 1858.

Read Also: Factorial formula

Key Terms: Cayley Hamilton Theorem, Square Matrices, Polynomial, Linear Algebra, Identity Matrix, Scalar Element.

---

What is Cayley Hamilton Theorem?

[Click Here for Sample Questions]

Cayley Hamilton Theorem determines that every square matrix over a commutative ring (including the real or complex field) agrees with its equation. Let's assume A as n×n matrix and In as identity matrix, then the distinctive polynomial of A will be expressed as

 P(x) = det (xl_n - A)

A polynomial m(x) is the minimal polynomial of A if

(i) m(A) = 0

(ii) m(x) is a monic polynomial (the coefficient of the highest degree term is 1)

(iii) if a polynomial g(x) is such that g(A) = 0, then m(x) divides g(x).

Read Also: 

---

Cayley Hamilton Theorem Statement 

[Click Here for Sample Questions]

The characteristic polynomial p(λ) = det(λ_In−A) can be decomposed as p(λ) = a_nλ_n+a_n−1λ_n−1+...+a₁λ + a₀λ₀.

This is a monic polynomial, here the leading coefficient, i.e, the coefficient of the highest degree variable, will be equal to 1.

Thus, a_n= 1. Here, a_n−1,...,a₁, a₀ are coefficients of the variables λ_n−1,..., λ₁, λ₀ respectively.

Read More: Concept of Elementary Row and Column Operations

---

Cayley Hamilton Theorem Formula

[Click Here for Sample Questions]

The formula of Cayley Hamilton Theorem formula is helpful in solving complicated and complex calculations and that too with accuracy and speed. Cayley Hamilton Theorem is also used to find out the inverse of a matrix. The formula is given as follows:

Suppose the characteristic polynomial of an n × n square matrix, A, is given as 

p(λ) = λⁿ +a_n−1λⁿ⁻¹+...+a₁λ+a₀

Then p(A) = Aⁿ+a_n−1Aⁿ⁻¹+...+a₁A+a₀In= 0

Thus, p(A) = 0.

In order to find the inverse, multiply this equation with A^-1.

Aⁿ⁻¹+a_n−1Aⁿ⁻²+...+a₁In+a₀A⁻¹= 0

A^-1 = -(Aⁿ⁻¹+a_n−1Aⁿ⁻²+...+a₁In) divide by a₀.

\(A^{-1} = - \frac{A^{n-1}A^{n-2}+...+a_1ln+\ a_0A^{-1}}{a_0}\)

Check Also: Statistics

---

Cayley Hamilton Theorem 2 × 2 

[Click Here for Sample Questions]

When you apply Cayley Hamilton Theorem to a 2 × 2 square matrix, the initial step is to find the characteristic polynomial expression. Characteristic polynomial general form is:

p(λ) = λⁿ + a_n−1λⁿ⁻¹ +...+ a₁λ + a₀λⁿ + a_n−1λⁿ⁻¹ +...+ a₁λ + a₀

As n = 2,

thus,

p(λ) = λ² +a₁λ + a₀

For a 2 × 2 square matrix this polynomial is written as

p(λ) = λ²−S₁λ +S₀

where, S₁ = sum of the diagonal elements and S0 = determinant of the 2 × 2 square matrix.

Now according to the Cayley Hamilton theorem, if λ is substituted with a square matrix then the characteristic polynomial will be 0. The formula can be written as

B₂−S₁B+S₀I = 0

In the above equation, B is a 2 × 2 square matrix.

Read Further: Multiplication and Division of Integers

---

Cayley Hamilton Theorem 3 × 3 

[Click Here for Sample Questions]

For a 3 × 3 square matrix the characteristic polynomial is given as,

 p(λ₃−T₂λ₂+T₁λ−T₀)

where,

T₂ = sum of the main diagonal elements,

T₁ = sum of the minors of the main diagonal elements,

T₀ = determinant of the 3 × 3 square matrix.

When you apply the Cayley Hamilton theorem, the formula is written as: C₃−T₂C₂+T₁C−T₀I = 0

In this equation, C is a 3 × 3 square matrix.

---

Points to Remember 

[Click Here for Sample Questions]

• Cayley Hamilton Theorem defines important concepts in control theory like the controllability of linear systems.
• According to Cayley Hamilton Theorem, all the square matrix satisfies its characteristic equation.
• Cayley Hamilton Theorem is only applicable for square matrix.
• The theory is named after mathematicians Arthur Cayley and William Rowan Hamilton.
• Cayley Hamilton Theorem was invented in the year 1858.

Check PYQs: 

---