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Let u,v <- C^n and set A := I + uv^H <- C^nxn.

(a) Suppose A is invertible. Prove that Inverse(A) = I_n + auv^H , for some a <- C. Give an expression for a.
(b) For what u and v is A singular?
(c) Suppose A is singular. What is the null space of A, N(A) in this case?

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This solution shows how to solve three basic algebra problems.

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