Linear Algebra and Calculus

Q1. Find the Eigenvalues and corresponding Eigenvectors for the matrix A = [[1, 2, 2], [2, 1, 2], [2, 2, 1]]. Verify the Cayley-Hamilton Theorem and find A^-1.

Key Points: Compute det(A - lambda*I) = -(lambda - 5)(lambda + 1)^2 = 0. Eigenvalues: 5, -1, -1. Solve (A - lambda*I)X = 0 to get eigenvectors. State Cayley-Hamilton A^3 - 3A^2 - 9A - 5I = 0. Multiply by A^-1 to solve for A^-1.

Q2. State and prove Euler’s Theorem for a homogeneous function u = f(x, y) of degree n. Hence evaluate x(du/dx) + y(du/dy) for u = sin^-1((x^2 + y^2)/(x + y)).

Key Points: Proof: Let u = x^n * phi(y/x). Differentiate with respect to x and y, combine x(du/dx) + y(du/dy) = n*u. For sin(u) = (x^2+y^2)/(x+y) (homogeneous degree 1), result is 1 * tan(u).

Q3. Expand f(x, y) = e^x * cos(y) in powers of (x - 1) and (y - pi/4) up to second-degree terms using Taylor’s Theorem for two variables.

Key Points: Calculate f(a,b), fx, fy, fxx, fxy, fyy at (1, pi/4). Plug into Taylor formula f(a+h, b+k) = f(a,b) + [h fx + k fy] + (1/2)[h^2 fxx + 2hk fxy + k^2 fyy].

End Semester Exam Dec 2025 / Jan 2026

End Semester Exam 2024