Consider the two statements given below: I. There exists a matrix N ∈ M4(ℝ) such that {(1, 1, 1, -1), (1, -1, 1, 1)} is a basis of Row(N) and (2, 1, 1, 4) ∈ Null(N). II. There exists a matrix M ∈ M4(ℝ) such that {(1, 1, 1, 0)T, (1, 0, 1, 1)T} is a basis of Col(M) and (1, 1, 1, 1)T, (1, 0, 1, 0)T ∈ Null(M). Which of the following options is true?

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?