?Assuming that the stated inverses exist, prove the following equalities.a

Chapter 1, Problem 24

(choose chapter or problem)

Assuming that the stated inverses exist, prove the following equalities.

a. \(\left(C^{-1}+D^{-1}\right)^{-1}=C(C+D)^{-1} D\)

b. \((I+C D)^{-1} C=C(I+D C)^{-1}\)

c. \(\left(C+D D^{T}\right)^{-1} D=C^{-1} D\left(I+D^{T} C^{-1} D\right)^{-1}\)

Partitioned matrices can be multiplied by the row-column rule just as if the matrix entries were numbers provided that the sizes of all matrices are such that the necessary operations can be performed. Thus, for example, if 𝐴 is partitioned into a 2 × 2 matrix and 𝐵 into a 2 × 1 matrix, then

\(\mathrm{AB}=\left[\begin{array}{ll} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right]\left[\begin{array}{l} B_{1} \\ B_{2} \end{array}\right]=\left[\begin{array}{l} A_{11} B_{1}+A_{12} B_{2} \\ A_{21} B_{1}+A_{22} B_{2} \end{array}\right] \)

provided that the sizes are such that 𝐴𝐵, the two sums, and the four products are all defined.

Equation Transcription:

AB=[][] = []

Text Transcription:

(C^-1+D^-1)^-1=C(C+D)^-1 D

(I+CD)^-1C=C(I+DC)^-1

(C+DDT)^-1D=C^-1D(I+DTC^-1D)^-1

AB=[A_21  A_22 A_11 A_12][B_2B_1] = [A_21B_1+ A_22B_2A_11B_1+ A_12B_2]