How to find the right balance between a challenging and an enjoyable mathematics exercise session?
As a grad student now who took undergrad linear algebra with a professor I really liked, I will attempt to generalize what I saw that worked.

The best way for students to learn something that is as potentially abstract as linear algebra is to relate it to something concrete. My professor always liked to point out that once you leave R2 and R3, everyone visualizes geometry differently. As a result, he often tried to keep his visual explanations in that space.

However, for higher dimensional vectors, you can still relate the material to multi-dimensional concepts without drawing a picture. A great example (and pretty interesting exercise) is Markov chains. When we went over eigenvalues and eigenvectors, we motivated all of the math by walking through Markov chains. He used Markov chain gambling schemes (e.g. holding a bet steady when you win, doubling when you lose) to illustrate eigen-decomposition in a practical way. In that way you can help students develop intuition by talking about something that makes sense outside of the math-sphere.

In terms of practical exercises, I've found that progressive problems tend  to work best. That is, where each problem is working towards one ultimate problem solution. Generally, each step is a little more complex as well. For example, maybe the first step is finding the Eigenvalues and Eigenvectors, the second is diagonalization, the third is something tangential like inverting the matrix, and the last is applying what you've calculated to some practical problem like a Markov Chain.

In other words, ease students into the harder material by showing them the simpler ways to do it first. Once you can intuitively understand the general concept at basic level, it's easier to move on to more advanced views of the same and similar concepts. Plus, when students see complex material from the viewpoint of something simpler to understand (like maximizing gambling winnings), it provides a reference point for them to start from.