I like Gilbert Strang’s Introduction to Linear Algebra (there are videos too: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...). The book has answers for some, but not all, of the problems.

I think Strang does a good job balancing intuition and rigor, without too much mindless grinding. Other books I’ve used either leap right into “Let F be a field....” or have you learn a bunch of matrix manipulations but don’t really explain why these are interesting or useful.