Prove that if matrices ABC are nxn and ABC J then B is inve

Prove that if matrices A,B,C are nxn and ABC = J then B is invertible and B = CA. Justify each step in your proof.

Solution

if A,B,C are nxn matrices then ABC is an nxn matrix. and for the matrix ABC to be invertible d e t ( A B C ) 0 det(ABC)=det(A)*det(B)*det(C) and det(ABC)0

Now ABC = I (Given)

Multiply both sides by C^-1 : ABCC^-1 = C^-1

AB = C^-1

ABB^-1 = C^-1B^-1

A = C^-1B^-1

Now multiply both side by C

CA =CC^-1B^-1 = B^-1

CA = B^-1