Inverse of an Elementary Matrix

Question

Assume we have a 3x3 matrix like:

A =  9 8 7
     6 5 4
     3 2 1

We are applying an Elementary matrix E to A:

E = 1 0 0
    0 1 0
    0 3 1

which gives A':

A' = 9  8  7
     6  5  4
     21 17 13

I need to find an E' that satisfies (E')*(A') = (A) .I tried below method but I'm not sure if it is correct or not:

E*A = A' => Let's multiply both side with inverse of E:
(E^-1)*E*A = (E^-1)*A' ; (E^-1)*E = I ; I*A = A;
A = (E^-1)*A'

E is type 3 elementary matrix. So inverse of E:

(E^-1) = 1 0 0
         0 1 0
         0 -3 1 

which is equal to E'. My question is, is this method correct, especially multiplying both sides with inverse of E.

Thanks in advance.

Answer 1

Your method is valid but one can also say $$E'A'=A$$ $$E'EA=A$$ $$E'E=I$$ $$E'=E^{-1}$$ $E'E=I$ because the identity matrix satisfies $IA=A$.