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## Homework Statement

If A, B, and C are nxn matrices, with B and C nonsingular, and b is an n-vector, how would you implement the formula

x = B[tex]^{-1}[/tex] (2A + I) (C[tex]^{-1}[/tex] + A)b

without computing any matrix inverses?

## Homework Equations

Is there any identity for (2A+I)[tex]^{-1}[/tex] that is expressed without the inverse?

## The Attempt at a Solution

x = B[tex]^{-1}[/tex] (2A + I) (C[tex]^{-1}[/tex] + A)b

Bx = (2A + I)(C[tex]^{-1}[/tex] + A)b

(2A + I)[tex]^{-1}[/tex]Bx = C[tex]^{-1}[/tex]b + Ab

If (2A+I)[tex]^{-1}[/tex] is expressed without the inverse, I would have proceeded as follows:

(2A + I)[tex]^{-1}[/tex]Bx - Ab = C[tex]^{-1}[/tex]b

C[(2A + I)[tex]^{-1}[/tex]Bx - Ab] = CC[tex]^{-1}[/tex]b

C[(2A + I)[tex]^{-1}[/tex]Bx - Ab] = b

C(2A + I)[tex]^{-1}[/tex]Bx - CAb = b

C(2A + I)[tex]^{-1}[/tex]Bx = (CA+I)b