'Taking transpose of transpose matrix gives the initial matrix'.
Is there an interesting real life example that can be used to explain this concept. Thanks in advance for any reply.
I'm not sure what "real world" means for you. You could think of it as flipping the matrix across the main diagonal, in which case it's geometrically intuitive that flipping it again gets you back where you started. I think of transposes as induced maps on dual spaces, but I don't know of any real-world analogies for that concept.
and what happens when we do this twice? think of a typical entry ai,j in the matrix A.
when we take the transpose, it is now in position j,i. when we take the transpose again, it goes to position i,j again...back where it started.
if this argument does not convince you, practice with a few 3x3 matrices.