[3 points]. Prove the following matrix identity
(P −1 + BT R−1B)−1BT R−1 = P BT (BP BT + R)−1, (1)
where P ∈ RN ×N , B ∈ RM ×N , and R ∈ RM ×M . P and R are invertible.
Note that if M N , it will be much cheaper to evaluate the right-hand side
than the left-hand side. Using Eq. (1), prove a special case
(I + AB)−1A = A(I + BA)−1,
where A ∈ RN ×M and B ∈ RM ×N