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[Community Question] Linear-algebra: Number of possible zero entries in orthogonal matrices

One of our user asked:

It's easy to check that in orthogonal matrix dimension $2 \times 2$ if there is entry $0$ in the matrix necessary one additional zero must be present. Then the total number of zeros is $2$.

In an orthogonal matrix dim. $3 \times 3$ number of zeros can be (if they are present) , I suppose from observations, only $4$ or $6$ - once again we obtain an even number of possible zeros.

  • Can this observation be extended for other orthogonal matrices of greater dimensions? The number of zeros is always even? How to prove this?

  • Maybe, it is known the explicit formula for the number of possible zeros in orthogonal matrices of any dimension?


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