Does an oriented graph contain circuits if and only if at least one element of the main diagonal is 1?

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In a course PDF the following proposition is written:

An oriented graph contains circuits if and only if at least one element of the main diagonal is 1.

I thought of an example that contradicts this proposition:

simple graph example

How can the quoted proposition be true?

Update: Is the proposition about the connections matrix and not about the adjacency matrix as I thought first?