Multiple choice

Given an orthogonal matrix $\begin{bmatrix} \ 1 & 1 & 1 & 1 \ \ 1 & 1 & -1 & -1 \ \ 1 & -1 & 0 & 0 \ \ 0 & 0 & 1 & -1 \end{bmatrix}$$\begin{bmatrix} \ AA^T \ \end{bmatrix}^{-1}$ is

  1. $\begin{bmatrix} \ \frac{1}{4} & 0 & 0 & 0 \\ \ 0 & \frac{1}{4} & 0 & 0 \\ \ 0 & 0 & \frac{1}{2} & 0 \\ \ 0 & 0 & 0 & \frac{1}{2} \\ \end{bmatrix}$
  2. $\begin{bmatrix} \ \frac{1}{2} & 0 & 0 & 0 \\ \ 0 & \frac{1}{2} & 0 & 0 \\ \ 0 & 0 & \frac{1}{2} & 0 \\ \ 0 & 0 & 0 & \frac{1}{2} \\ \end{bmatrix}$
  3. $\begin{bmatrix} \ 1 & 0 & 0 & 0 \\ \ 0 & 1 & 0 & 0 \\ \ 0 & 0 & 1 & 0 \\ \ 0 & 0 & 0 & 1 \\ \end{bmatrix}$
  4. $\begin{bmatrix} \ \frac{1}{4} & 0 & 0 & 0 \\ \ 0 & \frac{1}{4} & 0 & 0 \\ \ 0 & 0 & \frac{1}{4} & 0 \\ \ 0 & 0 & 0 & \frac{1}{4} \\ \end{bmatrix}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\text{From orthogonal matrix} \\ \begin{bmatrix} \ AA^T \ \end{bmatrix} = I \\ \text{Since the inverse of I is I, thus} \\ \begin{bmatrix} \ AA^T \ \end{bmatrix} = I^{-1} = I $

Keep practicing — related questions