[ubuntu_mate] 16.04 Installation of GRUB fails,

You have installed in the well known but 35 year old BIOS with MBR partitioning.
New systems use the newer UEFI with gpt partitioning.

You can stay with MBR/BIOS, but if you want to change now is the time.

Script says this:
SecureBoot maybe enabled.

New UEFI systems have three ways to boot, and how you boot installer is then how it installs. But how you boot the install or default settings in UEFI may not match the way you installed.
You can have UEFI with SecureBoot on, Standard UEFI and CSM.
CSM - UEFI Compatibility Support Module (CSM), which emulates a BIOS mode
Secure boot may just be called "Windows" and "Other" in UEFI.

You might try tuing Secure boot off in UEFI and see if grub then installs. Make sure in Boot-Repair you include separate /boot and LVM options.

You can also use gpt partitioning with BIOS boot. I converted to using gpt with my old BIOS system about 6 years ago. With BIOS you need a tiny bios_grub partition. But with UEFI you need the ESP - efi system partition. Even before I knew I was eventually going to get a new UEFI system, I started adding both the ESP & bios_grub to all new drives and even large flash drives. Then I would not have to totally refomat drive to add ESP as UEFI suggests it be first or at least near begiing of drive.

With LVM and encryption you need to lea about those. I do not know either.
And since not standard partitioning, and repairs of encrypted partitions can be very difficult extremely good backup plans are required.

GPT Advantages (older but still valid) see post#2 by srs5694:
http://ubuntuforums.org/showthread.php?t=1457901
https://wiki.archlinux.org/index.php...antages_of_GPT
http://en.wikipedia.org/wiki/Unified...ware_Interface
http://askubuntu.com/questions/62947...br-why-not-mbr
UEFI Advantages
http://askubuntu.com/questions/64730.../647604#647604
http://askubuntu.com/questions/44696...y-vs-uefi-help