Construct a space $X$ with $H_{0} ( X )=\mathbb{Z}, H_{1} ( X )=\mathbb{Z}_{2} \times\mathbb{Z}_{3}, H_{2} ( X )=\mathbb{Z},$ and all other homology groups of $X$ vanishing.

Options:
A. $$
A=\left( \begin{matrix} {{{1}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{1}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{-1}}} \\ \end{matrix} \right), \qquad B=\left( \begin{matrix} {{{\operatorname{c o s} \frac{\pi} {4}}}} & {{{\operatorname{s i n} \frac{\pi} {4}}}} & {{{0}}} \\ {{{-\operatorname{s i n} \frac{\pi} {4}}}} & {{{\operatorname{c o s} \frac{\pi} {4}}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right)
$$
B. $$
A=\left( \begin{matrix} {{{1}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{-1}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right), \qquad B=\left( \begin{matrix} {{{\operatorname{c o s} \frac{\pi} {8}}}} & {{{\operatorname{s i n} \frac{\pi} {8}}}} & {{{0}}} \\ {{{-\operatorname{s i n} \frac{\pi} {8}}}} & {{{\operatorname{c o s} \frac{\pi} {8}}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right)
$$
C. $$
A=\left( \begin{matrix} {{{2}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{2}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{-1}}} \\ \end{matrix} \right), \qquad B=\left( \begin{matrix} {{{\operatorname{c o s} \frac{5 \pi} {6}}}} & {{{\operatorname{s i n} \frac{5 \pi} {6}}}} & {{{0}}} \\ {{{-\operatorname{s i n} \frac{5 \pi} {6}}}} & {{{\operatorname{c o s} \frac{5 \pi} {6}}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right)
$$
D. $$
A=\left( \begin{matrix} {{{-2}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{-2}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right), \qquad B=\left( \begin{matrix} {{{\operatorname{c o s} \frac{\pi} {2}}}} & {{{\operatorname{s i n} \frac{\pi} {2}}}} & {{{0}}} \\ {{{-\operatorname{s i n} \frac{\pi} {2}}}} & {{{\operatorname{c o s} \frac{\pi} {2}}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right)
$$
E. $$
A=\left( \begin{matrix} {{{-1}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{-1}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right), \qquad B=\left( \begin{matrix} {{{\operatorname{c o s} \frac{2 \pi} {3}}}} & {{{\operatorname{s i n} \frac{2 \pi} {3}}}} & {{{0}}} \\ {{{-\operatorname{s i n} \frac{2 \pi} {3}}}} & {{{\operatorname{c o s} \frac{2 \pi} {3}}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right)
$$
F. $$
A=\left( \begin{matrix} {{{0}}} & {{{1}}} & {{{0}}} \\ {{{-1}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right), \qquad B=\left( \begin{matrix} {{{\operatorname{c o s} \frac{\pi} {6}}}} & {{{\operatorname{s i n} \frac{\pi} {6}}}} & {{{0}}} \\ {{{-\operatorname{s i n} \frac{\pi} {6}}}} & {{{\operatorname{c o s} \frac{\pi} {6}}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right)
$$
G. $$
A=\left( \begin{matrix} {{{0}}} & {{{1}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ {{{-1}}} & {{{0}}} & {{{0}}} \\ \end{matrix} \right), \qquad B=\left( \begin{matrix} {{{\operatorname{c o s} \frac{2 \pi} {3}}}} & {{{-\operatorname{s i n} \frac{2 \pi} {3}}}} & {{{0}}} \\ {{{\operatorname{s i n} \frac{2 \pi} {3}}}} & {{{\operatorname{c o s} \frac{2 \pi} {3}}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right)
$$
H. $$
A=\left( \begin{matrix} {{{0}}} & {{{0}}} & {{{1}}} \\ {{{0}}} & {{{-1}}} & {{{0}}} \\ {{{1}}} & {{{0}}} & {{{0}}} \\ \end{matrix} \right), \qquad B=\left( \begin{matrix} {{{\operatorname{c o s} \frac{7 \pi} {6}}}} & {{{\operatorname{s i n} \frac{7 \pi} {6}}}} & {{{0}}} \\ {{{-\operatorname{s i n} \frac{7 \pi} {6}}}} & {{{\operatorname{c o s} \frac{7 \pi} {6}}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right)
$$
I. $$
A=\left( \begin{matrix} {{{0}}} & {{{-1}}} & {{{0}}} \\ {{{1}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right), \qquad B=\left( \begin{matrix} {{{\operatorname{c o s} \frac{3 \pi} {4}}}} & {{{\operatorname{s i n} \frac{3 \pi} {4}}}} & {{{0}}} \\ {{{-\operatorname{s i n} \frac{3 \pi} {4}}}} & {{{\operatorname{c o s} \frac{3 \pi} {4}}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right)
$$
J. $$
A=\left( \begin{matrix} {{{-1}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{-1}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{-1}}} \\ \end{matrix} \right), \qquad B=\left( \begin{matrix} {{{\operatorname{c o s} \frac{\pi} {3}}}} & {{{\operatorname{s i n} \frac{\pi} {3}}}} & {{{0}}} \\ {{{-\operatorname{s i n} \frac{\pi} {3}}}} & {{{\operatorname{c o s} \frac{\pi} {3}}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right)
$$
Answer: E