Cayley-Hamilton Theorem – Engineering Maths for GATE, ESE, AE/JE | Priyanka Ma’am

The Cayley-Hamilton Theorem is one of the most important concepts in Linear Algebra. It is frequently used in GATE, ESE, AE/JE exams to find higher powers of matrices and inverse easily.

Statement of Cayley-Hamilton Theorem

Every square matrix satisfies its own characteristic equation.

If the characteristic equation of matrix A is:

λⁿ + a₁λⁿ⁻¹ + ... + aₙ = 0

Then:

Aⁿ + a₁Aⁿ⁻¹ + ... + aₙI = 0

Characteristic Equation (2×2 Matrix)

If A =

Characteristic equation:

λ² − (a + d)λ + (ad − bc) = 0

i.e., λ² − (Trace)λ + (Determinant) = 0

Using Cayley-Hamilton Theorem

Replace λ with matrix A:

A² − (Trace)A + (Determinant)I = 0

Applications (VERY IMPORTANT)

• Finding A², A³, Aⁿ
• Finding inverse of matrix
• Reducing higher powers

Solved Examples

Example 1

Verify Cayley-Hamilton for:

Trace = 4, Determinant = 3

Characteristic equation:

λ² − 4λ + 3 = 0

By Cayley-Hamilton:

A² − 4A + 3I = 0

(Verified)

Example 2 (Find A²)

Using:

A² = 4A − 3I

This avoids direct multiplication.

Example 3 (Find A⁻¹)

From:

A² − 4A + 3I = 0

Multiply both sides by A⁻¹:

A − 4I + 3A⁻¹ = 0

⇒ A⁻¹ = (4I − A) / 3

Example 4

If A satisfies:

A² − 5A + 6I = 0

Find A³

A² = 5A − 6I
A³ = A(5A − 6I) = 5A² − 6A
= 5(5A − 6I) − 6A
= 25A − 30I − 6A
= 19A − 30I

Shortcut Tricks

• Use trace & determinant directly for 2×2 matrix
• Avoid direct multiplication of matrices
• Always reduce higher powers using CH theorem
• Useful for inverse calculation

Questions

Q1. What does Cayley-Hamilton theorem state?

Answer: Matrix satisfies its own characteristic equation

Q2. Use of CH theorem?

Answer: Finding Aⁿ and inverse

Exam Tips

• Memorize 2×2 characteristic equation
• Use CH instead of multiplication
• Practice PYQs
• Avoid algebra mistakes

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