Example 42.68.23. Let $R = k$ be a field. Let $M = k^{\oplus 4}$. Let

\[ \varphi = \left( \begin{matrix} 0
& 0
& 0
& 0
\\ u_1
& 0
& 0
& 0
\\ 0
& 0
& 0
& 0
\\ 0
& 0
& u_2
& 0
\end{matrix} \right) \quad \varphi = \left( \begin{matrix} 0
& 0
& 0
& 0
\\ 0
& 0
& v_2
& 0
\\ 0
& 0
& 0
& 0
\\ v_1
& 0
& 0
& 0
\end{matrix} \right) \quad \]

with $u_1, u_2, v_1, v_2 \in k^*$. Then we have

\[ \det \nolimits _ k(M, \varphi , \psi ) = -\frac{u_1u_2}{v_1v_2}. \]

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