Consider the following series of square matrices:
\[
\begin{array}{l}
H_{1}=[1], \\
H_{2}=\left[\begin{array}{cc}
1 & 1 \\
1 & -1
\end{array}\right],
\end{array}
\]
and for $k=2,3, \ldots$, the $2^{k} \times 2^{k}$ matrix $H_{2^{k}}$ is recursively defined as
\[
H_{2^{k}}=\left[\begin{array}{cc}
H_{2^{k-1}} & H_{2^{k-1}} \\
H_{2^{k-1}} & -H_{2^{k-1}}
\end{array}\right] .
\]
What is $\left|\operatorname{det}\left(H_{2^{k}}\right)\right|?\; ($Hint: What is $H_{2^{k}} H_{2^{k}}^{T}?)$
- $0$
- $2^{k}$
- $2^{k / 2}$
- $2^{k 2^{k-1}}$
- $2^{k 2^{k}}$