Example 42.68.21. Let $R = k$ be a field. Let $M = k^{\oplus a} \oplus k^{\oplus b}$ be $l = a + b$ dimensional. Let $\varphi $ and $\psi $ be the following diagonal matrices
\[ \varphi = \text{diag}(u_1, \ldots , u_ a, 0, \ldots , 0), \quad \psi = \text{diag}(0, \ldots , 0, v_1, \ldots , v_ b) \]
with $u_ i, v_ j \in k^*$. In this case we have
\[ \det \nolimits _ k(M, \varphi , \psi ) = \frac{u_1 \ldots u_ a}{v_1 \ldots v_ b}. \]
This can be seen by a direct computation or by computing in case $l = 1$ and using the additivity of Lemma 42.68.18.
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