Proof: From the identity

(A + B)-1 = [A (A-1 + B-1) B]-1 = B-1 (A-1 + B-1)-1 A-1 or

(A + B)-1 = [B (B-1 + A-1)A]-1 = A-1 (A-1 + B-1)-1B-1.

Theorem 2: Let A and B be n x n PD matrices. If A + B is PD, then I + AB-1 and I + BA-1 are PD and

Proof: From the identity

-B-1(A-1 + B-1)-1B-1 = B-1 - B-1[A-1(I + AB-1)]-1B-1 = B-1 - B-1(I + AB-1)-1AB-1.

The left-hand side of this equation is (A + B)-1 (from Theorem 1). Hence,

Theorem 3: If A and B are PD matrices of order n and m, respectively, and if C is of order n x to, then

provided the inverse exists. Proof: From the identity

we have

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