Question

You are running a computer code with given matrices A, B, and C that are all size n × n and invertible. You put the code into a subroutine to calculate the inverse of ABC, but the subroutine gives an error indicating the matrix you gave it does not have an inverse. Show that the subroutine (written by someone else, not you) must be incorrect by showing that ABC is invertible by finding a matrix D so that (ABC)D = I and D(ABC) = I.

Answer 1

to prove

(ABC)D = I and D(ABC) = I

given : A,B,C are invertible

let their inverse are : A^-1, B^-1, C^-1

(ABC)D=I

multiply both sides by A^-1

A^-1(ABC)D=A^-1I

I(BC)D=A^-1

multiply both sides by B^-1

B^-1(BC)D=B^-1A^-1

I(C)D=B^-1A^-1

multiply both sides by C^-1

C^-1(C)D=C^-1B^-1A^-1

D=C^-1B^-1A^-1

similarly for D(ABC) = I

post multiply both sides by C^-1

D(ABC)C^-1 = IC^-1

D(AB)=C^-1

post multiply both sides by B^-1

D(AB)B^-1=C^-1B^-1

D(A)=C^-1B^-1

post multiply both sides by A^-1

D(A)A^-1=C^-1B^-1A^-1

D=C^-1B^-1A^-1

thus ABC is invertible and inverse is C^-1B^-1A^-1