Example 42.68.22. Let $R = k$ be a field. Let $M = k^{\oplus a} \oplus k^{\oplus a}$ be $l = 2a$ dimensional. Let $\varphi $ and $\psi $ be the following block matrices
\[ \varphi = \left( \begin{matrix} 0
& U
\\ 0
& 0
\end{matrix} \right), \quad \psi = \left( \begin{matrix} 0
& V
\\ 0
& 0
\end{matrix} \right), \]
with $U, V \in \text{Mat}(a \times a, k)$ invertible. In this case we have
\[ \det \nolimits _ k(M, \varphi , \psi ) = (-1)^ a\frac{\det (U)}{\det (V)}. \]
This can be seen by a direct computation. The case $a = 1$ is similar to the computation in Example 42.68.19.
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