Suppose that A is an n x n matrix such that A(Transpose)A=I. Let x be any vector in R^n. Show that llAxll=llxll; that is, multiplication of x by A produces a vector Ax having the same length as x.

Relevant equations

Sqrt(x(transpose)x)=llxll

The attempt at a solution

So I started setting up the equation wth an n x n matrix...

l a b l =A

l c d l

l a c l= A (transpose)

l b d l

A(transpose)A= I=

l a^(2) +c^(2) ab+cd l

l ba+dc b^(2)+d^(2)l

then I let the vector x=

l x1 l

l x2 l

then from there I was going to use the distance formula on Ax

but I wasn't sure if doing the dot product of Ax is the same thing as matrix multiplication of Ax, and my second problem is that I am not sure how to take the square root of a 2x2 matrix if they are the same thing.

If someone can answer those to questions for me I think I can finish the problem.

Thank you.