27-02-2025 B | Engineering Mathematics | GATE & ESE

Key Concepts:

• Cayley-Hamilton Theorem: Every square matrix satisfies its characteristic equation.
• Characteristic Equation: An equation derived from a matrix that helps find eigenvalues.
• Matrix Polynomial: An expression involving a matrix raised to various powers.
• Eigenvalues: Special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots, characteristic values, proper values, or latent roots.
• Determinant: A scalar value that can be computed from the elements of a square matrix.
• Trace: The sum of the elements on the main diagonal of a square matrix.
• Limits: The value that a function "approaches" as the input "approaches" some value.

Cayley-Hamilton Theorem Explanation

The Cayley-Hamilton theorem states that every square matrix satisfies its characteristic equation. This means if you have a matrix A (n x n), it satisfies the equation det(A - λI) = 0, where λ is a scalar and I is the identity matrix. When you expand this determinant, you get a polynomial in λ. The theorem says you can replace λ with the matrix A itself, resulting in F(A) = null matrix. This is useful for finding higher powers of A or dealing with matrix polynomials.

Example Problem 1: Finding A³ using Cayley-Hamilton (Question 74)

1. Problem Statement: Given a 2x2 matrix A where trace(A) = a + d = 1 and det(A) = ad - bc = 1, find A³. The options are in terms of matrix polynomials.
2. Approach: Use the Cayley-Hamilton theorem to express A³ as a matrix polynomial.
3. Characteristic Equation: For a 2x2 matrix, the characteristic equation is λ² - (trace of A)λ + det(A) = 0. Substituting the given values, we get λ² - λ + 1 = 0.
4. Applying Cayley-Hamilton: Replace λ with A: A² - A + I = 0, which implies A² = A - I.
5. Finding A³: Multiply both sides of A² = A - I by A: A³ = A² - A.
6. Substitution: Substitute A² = A - I into the equation for A³: A³ = (A - I) - A = -I.
7. Answer: A³ = -I.
8. Additional Questions:
   • Find A⁻¹: Multiply A² - A + I = 0 by A⁻¹ to get A - I + A⁻¹ = 0, so A⁻¹ = I - A.
   • Find A⁹: A⁹ = (A³)³ = (-I)³ = -I.

Example Problem 2: Determinant of a Matrix Polynomial (Question 73)

1. Problem Statement: Given matrix A = [2 3; 2 5], find the determinant of A⁴ - 5A³ + 6A² + 2I.
2. Approach: Use the Cayley-Hamilton theorem to simplify the matrix polynomial.
3. Characteristic Equation: det(A - λI) = (2-λ)(5-λ) - 6 = λ² - 7λ + 4 = 0.
4. Applying Cayley-Hamilton: A² - 7A + 4I = 0.
5. Simplifying the Polynomial: Notice the coefficients 1, -5, and 6 in the given polynomial resemble those in the characteristic equation. Rewrite the polynomial as A²(A² - 5A + 6I) + 2I.
6. Using the Cayley-Hamilton Result: Since A² - 5A + 6I = 0, the expression simplifies to A²(0) + 2I = 2I.
7. Finding the Determinant: det(2I) = 2² * det(I) = 4 * 1 = 4.
8. Alternative Method (Using Eigenvalues):
   • Find the eigenvalues of A: λ₁ = 2, λ₂ = 3.
   • Find the eigenvalues of the polynomial matrix B = A⁴ - 5A³ + 6A² + 2I: λ_B = λ_A⁴ - 5λ_A³ + 6λ_A² + 2.
   • Calculate λ_B for each λ_A:
     • λ_A = 2: λ_B = 2⁴ - 5(2³) + 6(2²) + 2 = 2.
     • λ_A = 3: λ_B = 3⁴ - 5(3³) + 6(3²) + 2 = 2.
   • det(B) = product of eigenvalues of B = 2 * 2 = 4.

Example Problem 3: Matrix Polynomial and Characteristic Equation (2025 Electrical Gate Question)

1. Problem Statement: Given matrix P = [2 1 0; -1 0 0; 0 0 1], find P².
2. Approach: Find the characteristic equation and use the Cayley-Hamilton theorem.
3. Characteristic Equation: det(P - λI) = (1-λ)((2-λ)(-λ) + 1) = (1-λ)(λ² - 2λ + 1) = 0.
4. Applying Cayley-Hamilton: (I - P)(P² - 2P + I) = 0.
5. Simplifying: From the characteristic equation, either P = I or P² - 2P + I = 0.
6. Solving for P²: If P² - 2P + I = 0, then P² = 2P - I.
7. Answer: P² = 2P - I.

Example Problem 4: Finding A⁹ using Cayley-Hamilton and Eigenvalues (Question 72)

1. Problem Statement: Given A² + 3A + 2I = 0 and A⁹ = [ -3 2; -1 0], find A⁹⁹.
2. Approach: Use the Cayley-Hamilton theorem to find a pattern and then use eigenvalues to verify.
3. Applying Cayley-Hamilton: A² = -3A - 2I.
4. Finding A³: A³ = -3A² - 2A = -3(-3A - 2I) - 2A = 7A + 6I.
5. Pattern Recognition: Notice that the coefficients are consecutive (e.g., -3, -2 and 7, 6).
6. Generalization: A⁹ = 511A + 510I (deduced by observing the pattern).
7. Eigenvalue Verification:
   • Find eigenvalues of A: λ₁ = -1, λ₂ = -2.
   • Find eigenvalues of A⁹: λ₁⁹ = (-1)⁹ = -1, λ₂⁹ = (-2)⁹ = -512.
   • Check if the eigenvalues of 511A + 510I match:
     • For λ₁ = -1: 511(-1) + 510 = -1 (matches).
     • For λ₂ = -2: 511(-2) + 510 = -512 (matches).
8. Conclusion: A⁹⁹ = 511A + 510I.

Calculus Introduction

The discussion transitions to calculus, emphasizing that calculus terms like continuity, differentiability, radius of curvature, and integration are all limiting forms. The initial focus will be on understanding limits before delving into calculus applications.

Standard Limits

The lecture mentions standard limits as a foundation for calculus. The basic definition of a limit of a function at x = a is the value that the function approaches as x approaches a.