Given an orthogonal matrix $A = \begin{bmatrix} 1 & 1 & 1 & 1\\ 1 & 1 & -1 & -1\\ 1 & -1 & 0 & 0\\ 0 & 0 & -1 & -1 \end{bmatrix}$, then [AA']-1 is

(a). $\begin{bmatrix} \frac{1}{4} & 0 & 0 & 0\\ 0 & \frac{1}{4} & 0 & 0\\ 0 & 0 & \frac{1}{4} & 0\\ 0 & 0 & 0 & \frac{1}{4} \end{bmatrix}$                               (b).    $\begin{bmatrix} \frac{1}{2} & 0 & 0 & 0\\ 0 & \frac{1}{2} & 0 & 0\\ 0 & 0 & \frac{1}{2} & 0\\ 0 & 0 & 0 & \frac{1}{2} \end{bmatrix}$

(c). $\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}$                               (d).    $\begin{bmatrix} \frac{1}{4} & 0 & 0 & 0\\ 0 & \frac{1}{4} & 0 & 0\\ 0 & 0 & \frac{1}{2} & 0\\ 0 & 0 & 0 & \frac{1}{2} \end{bmatrix}$

Ans. In orthogonal matrix, $A^T=A^-1$

$AA^T=I$

$[AA^T]=[AA^-1]=I^-1=I$