Tutoring is depressing because my students are struggling too much with my exercises. What do I do?
What you're experiencing is pretty much what all college teachers are wrestling with all the time everywhere.
Allow me to point out that the abyss of need can go much, much deeper than what you're currently dealing with. In the U.S., most college students are attending a community college. And most students entering U.S. community colleges do not have 8th grade algebra skills, nor even 6th grade arithmetic skills (e.g., fractions, proportions, negatives, estimations, times tables). I've had a good chunk of a college math lecturer career at this point and I've never taught anything as high level as matrices.
Here's how I've been able to jiu-jitsu it in my own mind. Think of it like a diagnostician. Doctors have to find it fascinating to track down hideous diseases. Detectives have to find it compelling to decipher ostensibly horrible crime cases. So too: I find it endlessly fascinating to diagnose what exactly is the block or problem in students' heads, or to track down the place where their reasoning first went off the rails. Ask questions to try and find where their reasoning starts and where it stops. Do something of a binary search to try to narrow down the problem spot. I'm continually discovering new gaps in people's knowledge that I never would have expected. It's surprising and amazing every day.
Usually I do find that students simply cannot remember starting definitions. If they've come up through a system that depends on raw faith and memory (and not mathematical reasoning), they don't get how important that axiomatic basis is. Many also clearly have learning disabilities that make this difficult for them. "All your answers are back in the definitions", I say several dozen times per semester. Have the book handy so you can turn back to the starting definitions at any time, and see if they recall or can see how their reasoning got off track that way. In fact -- this explains why at your current school they're asking students to reiterate definitions; whereas for us that's natural and obvious, for some it's an overwhelmingly difficult blind spot to fill in.
It sounds to me that the course itself is designed to be a gentler introduction to some of the mathematics they will be encountering in their future careers. If the goal of the course is supposed to be difficult and demanding, but it is not being taught that way, that's a different question. However, it sounds instead that the actual course itself is not designed to be at that high a level.
Think about the learning process that everyone goes through. I can use the exact same token to say that a grade 1 math class isn't "rigorous" enough and that the students are not gaining the intuition I want them to. In this case, it's not appropriate to make the course more difficult since the children are gaining that intuition through repetition, simple problems, and exposure to the concepts.
In your case, I would think about the learning objectives of the course, and the level of the students, and their eventual careers. Then, I would try to figure out how I can get them to learn as much as they can against those learning objectives, without necessarily turning them into mathematicians if that isn't what they're there for.
It is very often students get tutors to help them with the tests. They aren't motivated by the need of knowledge, but by their GPA.
There are 2 possible strategies:
You help them with the kind of problems that could show up on the tests. I actually did this with my cousin who had a calculus exam. I explained her the homework problems, she memorized the solutions, she passed. She still doesn't know what a derivative is. In my naivety, I thought she simply understood one semester of calculus we tried to cram in one day. No one is that smart.
If your student has a little patience, you could help them work through problems, however trivial they appear to you. Your position is to help the student discover concepts for themselves. I do not master this art, but I remember primary school teachers who were good at it. What they do, they ask questions about different parts of the problem until it becomes clear what is the piece of knowledge the student is missing. For example, if you get a matrix equation, and they can't solve it, you ask if they know how to multiply matrices, what are matrices, until you get to the point of realization that they don't know what an equation is. As discouraged as you may be at that point, you can still get them to learn what an equation is, solve a few, and so on, until you succeed to get your student to solve the original problem. My hunch is that most students will hate you for this approach.
The point of this exercise is to create the network of facts and procedures the student needs to solve even basic problems, i.e. something to hang their thoughts to. You should avoid as much as possible solving things for them more than once. And you should never offer generalizations before they can think of a need for them.