Given an orthogonal matrix
$$A= \begin{pmatrix} 1 & 1 & 1 &1 \\ 1 & 1 & -1 &-1\\ 1 & -1 & 0 &0\\ 0 &0&1&-1 \end{pmatrix}, $$
then values of $[AA^{T}]^{-1}$ is ?
My question is how come it is an orthogonal matrix $1^2+1^2+1^2+1^2\ne1$
Sum of squares of rows should be one isnt it?
This is going to depend on your definitions. While most people require the rows to be orthonormal, some people will say that a matrix with orthogonal rows is "orthogonal." This matrix falls into that category. If you divide all the rows by their lengths, you will get an actual orthogonal matrix.
It is worth noticing that the columns here are NOT orthogonal, but they would be if you divide the rows by their lengths. Right now, you only have that $AA^T$ is diagonal. After normalizing, you would have that $AA^T=I$.