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

with $U, V \in \text{Mat}(a \times a, k)$ invertible. In this case we have

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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