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Start Free Sprint →Focus on Eigenvalues & Eigenvectors of 3x3 matrices, Cayley-Hamilton theorem, Taylor & Maclaurin series expansion in two variables, Partial derivatives (Euler’s theorem on homogeneous functions), and Gauss Elimination / Rank of a matrix.
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.
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).
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].
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