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Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample. i) The relation ∼ defined by R by x ∼ y if x ≥ y is an equivalence relation. ii) If S1 and S2 are finite non-empty subsets of a vector space V such that [S1] = [S2], then S1 and S2 have the same number of elements. iii) For any square matrix A, ρ(A) = det(A) iv) The determinant of any unitary matrix is 1. v) If the characteristic polynomials of two matrices are equal, their minimal polynomials are also equal. vi) If the determinant of a matrix is 0, the matrix is not diagonalisable. vii) Any set of mutually orthogonal vectors is linearly independent. viii) Any two real quadratic forms of the same rank are equivalent over R. ix) There is no system of linear equations over R that has exactly two solutions. x) If a square matrix A satisfies the equation A2 = A, then 0 and 1 are the eigenvalues of A.
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