For each of the following 18 statements, either leave a blank answer or state whether it is true or false. Correct answers will score 1 mark, incorrect answers will lose 1\frac{1}{2} marks, so don’t guess. Blank answers score zero. Negative total scores will be rounded up to zero. Full marks can be obtained for 10 correct answers with none incorrect. Higher scores will be rounded down to 10.

(a) The eigenvectors of a matrix are always mutually perpendicular.

(b) The modulus of the complex number 7 + i is strictly greater than that of 5 − 5i.

(c) When performing Gaussian Elimination, it is a good idea to use the pivot of smallest possible magnitude to prevent loss of accuracy.

(d) If \underline{a} ∧ (\underline{b} ∧ \underline{c}) = 0, then the three non-zero vectors \underline{a}, \underline{b} and \underline{c} must be coplanar.

(e) If, in addition to the conditions of (d), \underline{b}∧\underline{c} ≠ 0, then the three vectors cannot be coplanar.

Questions (f)-(m) relate to the matrices A and B, where

detB ≠ 0 and A = ( \begin{matrix}2 & 1 & 0 \\ 7 & 3 & 1 \\ 1 & 0 & 1 \end{matrix} ) .

(f) The matrix A^{−1} exists and has the same eigenvectors as A.

(g) A cannot be written as the product of rotation matrices and non-zero scaling matrices.

(h) The columns of A are linearly dependent.

(i) The sum of the eigenvalues of A is 6.

(j) The vector A^{n}\underline{x} tends to zero as n → ∞ for any vector \underline{x}.

(k) If B\underline{x} = \underline{p}, then \underline{x}^{T} = \underline{p}^{T} B^{−1}.

(l) B\underline{x} = \underline{p} has a unique solution for \underline{x}.

(m) (detB)^{−1} = det(B^{−1}).

(n) By de Moivre’s theorem, cos 3θ = ℜe[(cos θ + i sin θ)^{3}] = cos^{3} θ − 3 cos θ sin^{2} θ.

(o) The complex roots of the equation z^{4} + iz^{2} + z − 1 = 0 occur as complex conjugate pairs.

(p) If z_{1} and z_{2} are complex numbers, |z_{1} + iz_{2}| = \sqrt{z_{1}^{2} + z_{2}^{2}}.

(q) If a set of simultaneous linear equations has two solutions, then it must have a third.

(r) The surfaces defined by the position vector \underline{R} ≡ (x, y, z) satisfying \underline{R}·\underline{a} = 0 and |\underline{R}−\underline{a}| = 1 intersect in a circle of radius 1.