Did I calculate the inverse of this matrix correctly?

Question

I tried doing it in a different way than I normally do and I am not sure if it is correct. I looked it up here and I know that even if it looks different it may still be the same matrix but it's hard to tell. Did I do it right?

Another question: if I had done it right and took the determinant of the inverse matrix I got and put the original one in that website and taken the determinant of the resulting one would they be the same? In other words, is the determinant basically a way of checking if two matrices are the same even if they look different because elementary operations don't change this determinant?

Answer 1

Regarding your second question, if two matrices are equal, then they have the same determinant. But the fact that two matrices have the same determinant does not imply that the matrices are equal.

Answer 2

What you have done is wrong. When you use Gauss' method for obtaining the inverse matrix, you can only do operations with rows, not with columns (not even changing two columns, which is what you did.)