Let A be an nxn matrix that is similar to an upper triangular matrix and has the distinct eigenvalues 1 , 2 , , k with corresponding multiplicities m 1 , m 2 , , m k Prove the following statements (a) tr(A) i 1, k m i i (b) det(A) ( 1 )m 1 ( 2 )m 2 ( k )m k ...

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Let A be an nxn matrix that is similar to an upper triangular matrix and has the

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Question:

Let A be an nxn matrix that is similar to an upper triangular matrix and has the distinct eigenvalues λ₁, λ₂, ..., λ_k with corresponding multiplicities m₁, m₂, ..., m_k. Prove the following statements:

(a) tr(A) = ∑[i=1, k] m_iλ_i

(b) det(A) = (λ₁)m₁(λ₂)m₂... (λ_k)m_k