Let A be an n x n matrix whose characteristic polynomial is p() n a1 n 1 an 1 an If A is nonsingular, show that A 1 1 an (An 1 a1 An 2 an 2 A an 1 In)

Question: Let A be an n x n matrix whose characteristic polynomial is p() = n + a1 n-1 + .... + an-1 + an.

Let A be an n x n matrix whose characteristic polynomial is
p(λ) = λn + a1 λn-1 + .... + an-1 λ + an.
If A is nonsingular, show that
A-1 = - 1/an (An-1 + a1 An-2 + .... + an-2 A + an-1 In)?

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