We need to compute the element $a_{11}$ of the matrix $A = (Q^T Q)^3$, where $Q = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 2 & 0 \\ 2 & 0 & 0 \\ 0 & 0 & 2 \end{pmatrix}$.

1. Problem Description

We need to compute the element $a_{11}$ of the matrix $A = (Q^T Q)^3$, where

$Q = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 2 & 0 \\ 2 & 0 & 0 \\ 0 & 0 & 2 \end{pmatrix}$.

2. Solution Steps

First, we compute $Q^T Q$:

$Q^T = \begin{pmatrix} 0 & 0 & 2 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 2 \end{pmatrix}$.

$Q^T Q = \begin{pmatrix} 0 & 0 & 2 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 2 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 \\ 0 & 2 & 0 \\ 2 & 0 & 0 \\ 0 & 0 & 2 \end{pmatrix} = \begin{pmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{pmatrix} = 4I_3$,

where $I_3$ is the $3 \times 3$ identity matrix.

Then we have

$A = (Q^T Q)^3 = (4I_3)^3 = 4^3 I_3^3 = 64 I_3 = \begin{pmatrix} 64 & 0 & 0 \\ 0 & 64 & 0 \\ 0 & 0 & 64 \end{pmatrix}$.

The element $a_{11}$ is the element in the first row and first column of the matrix $A$.

3. Final Answer

$a_{11} = 64$.