Let f : ℝ3 → ℝ be a function defined by f(x) = x'Ax, where A is a 3 × 3 matrix with real entries and x' denotes the transpose of vector x. The gradient of f at a point x0 is necessarily given by:

Let A ∈ &mathbb;R7×7. Define S = { λ ∈ &mathbb;R : det(A - λI) = det(λI - A) }. Then which of the following is correct?

Consider the matrix A = 11α1-10β21-1γ0. Which of the following statements is correct?

Let f : P3(&mathbb;R) → P3(&mathbb;R) be a linear operator such that f(1) = 1 + 2x + 3x2, f(x) = x + 2x2, f(x2) = x2, and f(x3) = x3. Which of the following statements is NOT true?