One of things I noticed when I self study is when I go check my answers against the solutions, some of my answers seem to be way off. For example there was a question that went "show that for any nxn non singular matrix it is row equivalent." I happened to show a proof by induction but in the solutions it just drew a arbirtrary nxn matrix with 1's running down the diagonal( these entries are obviously the pivot positions). I'm not sure if this normal or not?Will I just get better by practicing and struggling? But I fear I won't be able to get any better than I am now. The way I approach each section of a book is I first read very carefully and reread a few times before I feel I have a good understanding. Then I go to the problems and try to do all of them on my own. About 3/4 of them I can do and end up with the correct answer. Of course I recheck my solution. Then I check my solutions against the solution manual but at times like I said earlier my answers are way off than what is written in the solutions. This happens mostly on problems that say "show this...". I try my best to go back and fix my answer but I just end up leaving it since I feel its not worth it since I already know the solution. I'm trying hard to make sure that I can get up to the point where I can answer all the problems correctly but it seems quite hard.