Answer
Since A$A^{T}$ = I, the matrix is orthogonal.
$A^{-1}$= $\begin{bmatrix}
.5 & .5 & .5 & .5\\
.5 & .5 & -.5 & -.5\\
-.5 & .5 & -.5 & .5\\
-.5 & .5 & .5 & -.5
\end{bmatrix}$
Work Step by Step
If a matrix is orthogonal, then A$A^{T}$ = I. If a matrix is orthogonal, then the inverse of the matrix is equal to $A^{T}$.
First, we determine if A$A^{T}$ = I.
A = $\begin{bmatrix}
.5 & .5 & -.5 & -.5\\
.5 & .5 & .5 & .5\\
.5 & -.5 & -.5 & .5\\
.5 & -.5 & .5 & -.5
\end{bmatrix}$ $A^{T}$ = $\begin{bmatrix}
.5 & .5 & .5 & .5\\
.5 & .5 & -.5 & -.5\\
-.5 & .5 & -.5 & .5\\
-.5 & .5 & .5 & -.5
\end{bmatrix}$
A$A^{T}$ = $\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
\end{bmatrix}$
Since A$A^{T}$ = I, the matrix is orthogonal.
Since the matrix is orthogonal. $A^{-1}$ = $A^{T}$, so
$A^{-1}$= $\begin{bmatrix}
.5 & .5 & .5 & .5\\
.5 & .5 & -.5 & -.5\\
-.5 & .5 & -.5 & .5\\
-.5 & .5 & .5 & -.5
\end{bmatrix}$
