Given an orthogonal matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 1&1&1&1 \\ 1&1&{ - 1}&{ - 1} \\ 1&{ - 1}&0&0 \\ 0&0&1&{ - 1} \end{array}} \right],\,{\left[ {{\text{A}}{{\text{A}}^{\text{T}}}} \right]^{ - 1}}\,{\text{is}}\]

A. \[\left[ {\begin{array}{*{20}{c}} {\frac{1}{4}}&0&0&0 \\ 0&{\frac{1}{4}}&0&0 \\ 0&0&{\frac{1}{2}}&0 \\ 0&0&0&{\frac{1}{2}} \end{array}} \right]\]

B. \[\left[ {\begin{array}{*{20}{c}} {\frac{1}{2}}&0&0&0 \\ 0&{\frac{1}{2}}&0&0 \\ 0&0&{\frac{1}{2}}&0 \\ 0&0&0&{\frac{1}{2}} \end{array}} \right]\]

C. \[\left[ {\begin{array}{*{20}{c}} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right]\]

D. \[\left[ {\begin{array}{*{20}{c}} {\frac{1}{4}}&0&0&0 \\ 0&{\frac{1}{4}}&0&0 \\ 0&0&{\frac{1}{4}}&0 \\ 0&0&0&{\frac{1}{4}} \end{array}} \right]\]

Answer: Option C