Inverse of an Elementary Matrix

inversematrices

Assume we have a 3×3 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.

Best Answer

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 $IA=A$.

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