This cannot be X-linked recessive because of the picture of the gel that has been shown.
If it were x-linked recessive, then person 6 would be completely clean from the gene mutation since he is male. If that were the case, he should only be showing two small bands at 600 bp and 400 bp since he has only one good X chromosome.
However, his gel column shows a 1,000 bp fragment, which means that one of his genes was not cut by the Nhe1 enzyme. This implies that that one gene is being vut (hence the 600 bp and 400 bp fragment) and one gene is uncut (hence the 1,000 bp fragment). Since person 6 has two genes, and males only have one X chromosome, this must be an autosomal disease.
This is also reinforced by person 5, who has the same column as person 6. Person 6 also shows an uncut and cut gene by Nhe1. Since their dad has the disease, and neither of the children have the disease, it must be recessive in nature.
Person 7 has two normal genes (hence the two thick bands at 600 bp and 400 bp) and person 8 has two recessive genes (hence the one thick band at 1,000 bp). Their parents must both have a recessive trait for this to happen.
Hope this helps! Let me know if anything is unclear