?Suppose that an invertible matrix A is partitioned as\(A=\left[\begin{array}{ll} A_{11}

Chapter 1, Problem 26

(choose chapter or problem)

Suppose that an invertible matrix A is partitioned as

\(A=\left[\begin{array}{ll} A_{11} A_{12} A_{21} A_{22} \end{array}\right] \)

Show that

\(A^{-1}=\left[\begin{array}{llll} B_{11} & B_{12} & B_{21} & B_{22} \end{array}\right] \)

Where

\(B_{11}=\left(A_{11}-A_{12} A_{22}^{-1} A_{21}\right)^{-1}, \quad B_{12}=-B_{11} A_{12} A_{22}^{-1}\)

\(B_{21}=-A_{22}^{-1} A_{21} B_{11}, \quad B_{22}=\left(A_{22}-A_{21} A_{11}^{-1} A_{12}\right)^{-1}\)

provided all the inverses in these formulas exist.

Equation Transcription:

   

     

Text Transcription:

A=[A_11 A_12 A_21 A_22]

A^-1=[B_11 B_12 B_21 B_22]

B_11=(A_11-A_12 A_22^-1 A_21)^-1,      B12=-B_11 A_12 A_22^-1  

B_21=-A_22^-1 A_21 B_11,       B_22=(A_22-A_21A_11^-1 A_12)^-1