Let A be a 3 × 3 matrix with real entries. Which of the following assertions is FALSE?

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?

Suppose A and B are similar 3 × 3 matrices with real entries. Which of the following statements is NOT necessarily correct?

Let T : ℝ5 → ℝ9 be an injective (one-one) linear map. Let S : ℝ9 → ℝ13 be a linear map such that rank(ST) = 5. Then, rank(S) can attain all the values in the set:

If -1, 1, and 3 are the distinct eigenvalues of a real symmetric matrix corresponding to a quadratic form, then its canonical form will be:

Let A = 300130003, B = -300-3, and C = 0010. The minimal polynomial of the block diagonal matrix M = diag(A, B, C) is:

The rank of the matrix A = 2-10210311-1-314-204 is equal to: