Sorry for your loss. To answer your question: Yes, but... Based on my estate planning and real estate experience, lenders do not like trusts, especially irrevocable trusts. Unfortunately, your brother as trustee is going to have to do some searching to see if he can find a lender who will allow you to proceed as such.
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As is stated above, most lenders will not consider lending to Irrevocable Trusts and some will consider it, but require a higher level of assets in the trust or other collateral for the loan than what an individual will generally have to post/provide to the lender.
The lenders that will consider lending to an irrevocable trust will look at the assets of the trust and in some situations you can offer to personally guarantee the mortgage or provide some other incentive for the lender to make the loan.
Ultimately you will both have to find a lender who will consider this and meet their requirements for such a loan. Because the options will be limited, you may not find it to be worthwhile to structure it in the manner in which you set up the question. Good luck with your search.