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, \(A^T=A^-1 \)

\(AA^T=I\)

\([AA^T]=[AA^-1]=I^-1=I \)

So, c is correct answer.

Aniruddha Humane @anih 11 Jan 2018 12:28 am

List all such possible properties beforehand please