Problem based on skew-symmetric matrix

Question

Problem 1: Let A be a skew-symmetric n x n-matrix with entries in R, i.e. A^T = -A.

(a) Prove that u^TAu = 0 for every u ∈ Rⁿ.

(b) Prove that I_n + A is an invertible matrix.

(c) Give an example of a skew-symmetric 2 x 2-matrix B with entries in C for which I₂ + B is not invertible.

Problem 2: Let two vectors x^→ = (x₁, x₂, ..., x_n) and y→ = (y₁, y₂, ...., y_n).

a) Provide definition of Orthogonality.

b) Prove that if x^→ and y^→ are mutually orthogonal, then they are linearly independent.

Problem 3: If G is a group and H is a subgroup of G, then H is a normal subgroup of G if ghg^-1 ∈ H for all g from the set of generators of G and for all h from the set of generators of H.

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