This is a common misconception and is not correct. A new Certification of Trust should be prepared noting that you are the Co-Trustee and the Trust will need an EIN. Also, the Trust should be reviewed at this time because often times the Trust document directs that the surviving spouse and or the Trustee need to identify the assets and transfer them into new Trusts as set forth in the original Trust. Even if all assets remain in the original Trust, after the death of one spouse, the basis in the assets often changes, which can affect taxes.
Therefore it is important at this time to have an Estate Planning Attorney familiar with Trust Administration review the Trust.