Welcome to the forum @Aquassassinator.
This is how I check SHA256SUMS.
1. Open MATE Terminal and type in sha256sum
followed by a space.
2. Drag the .iso file to MATE Terminal and release. This will cause MATE Terminal to add the correct path to the .iso file.
Example:
sha256sum '/home/steven/Downloads/ubuntu-mate-16.04.2-desktop-amd64.iso'
3. Depress the Enter key and wait for the output.
4. If you also downloaded the SHA256SUMS file, copy the hash string from MATE Terminal
Example:
39cb4d4069dd79a9104b8c1c5d0d4a5b009779eec55fafeceafcf43c7ebcaba4
5. Open the the SHA256SUMS file in Pluma (text editor), and use the search function to search for the the hash string. If a match is found, Pluma will highlight it in the SHA256SUMS file.
The yellow highlight makes it quick and easy to see if the the hash strings match. Thus eliminating the need to verify each character of the hash string.
Alternative Method
1. If you downloaded the SHA256SUMS file from the server, place that file and the .iso file in the same folder. The Downloads folder will work fine.
2. Open MATE Terminal and enter cd Downloads
3. Next in MATE Terminal enter the following one line script to cause Mate Terminal to check the SHA256SUMS file against the .iso file and verify that the hash is correct.
sha256sum -c <(grep ubuntu-mate-16.04.2-desktop-amd64.iso SHA256SUMS)
4. This should result in a output simular to the following example.
sha256sum -c <(grep ubuntu-mate-16.04.2-desktop-amd64.iso SHA256SUMS)
ubuntu-mate-16.04.2-desktop-amd64.iso: OK
You will need to change ubuntu-mate-16.04.2-desktop-amd64.iso to the name of you .iso file your checking.