The trace of a matrix is defined to be the sum of its diagonal matrix elements Tr (Ω) = ΣΩ 2 Show that i. Tr(ΩΛ) = Tr(ΛΩ

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The trace of a matrix is defined to be the sum of its diagonal matrix elements Tr (Ω) = ΣΩ 2 Show that i. Tr(ΩΛ) = Tr(ΛΩ), ii. Tr(ΩΛ0) = Tr(Λ0Ω) = Tr(0ΩΛ), where 2, A and are matrices of same dimension. Suppose is a linear transformation or equivalently linear operator on a n- dimensional linear vector space, V. Find out the matrix representation of Twith respect to the basis vectors of this vector space, {lei)}. Suppose a) is a generalised vector in the aforesaid vector space V, such that a) = Σ=1 aj|ei). Action of T on a) is defined as a') = Tla). Hence show that n a => Tijaj, where, a (ei a'). The Hamiltonian operator for a two-state system is given by H = a(|1) (1||2) (2| + |1) (2| + |2) (1), where a is a number with the dimension of energy. Find the energy eigenvalues and the corre- sponding energy eigenkets (as linear combination of 1) and 2)).