More fun with binary chess

Algebra Level 5

How many invertible symmetric \(8\times8\) matrices containing 9 ones and 55 zeros exist?

Going by our first three moves, it appears that the game of binary chess is played as follows: Each of the two players misreads the other's question and creates an interesting new question in the process. Let's keep it going ;)

Obtained by misreading this.

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