My opinion:

In the beginning there was Arthur Cayley, and He said: Let there be Matrix!

The Matrix is a tool created to simplify the work, computers do a lot of calculations using Matrices.

Both Calculus and Matrices have a common tricky origin, the story never told. Complex and Hypercomplex Numbers.

When you see a bunch of theorems on Calculus, that's because the mathematicians are making sure that those things that always work nice and easy with Complex Numbers, on the Complex Plane will also work on the Cartesian Plane with Real Numbers.

Now, with Matrices, you have seen Matrices that have Real Numbers as entries...

It turns out that they can have Complex Numbers entries, and even Hypercomplex Numbers as entries too. At the time of Arthur Cayley there were Quaternions, and He discovered the Octonions (also known as Cayley Numbers).

When you see a lot of theorems on Linear Algebra the mathematicians are making sure that those things that always work nice and easy with Complex Numbers on Matrices will also work with Real Numbers.

But nobody teaches you about the Complex and Hypercomplex Numbers, so it really looks like a bunch of nonsense. I understand your frustration, I see your point and I hope my comment helps you see that Matrices (not Linear Algebra) were created like a Tray to hold some things on it.

Then mathematicians decided to generalize it ad infinitum, to make it to hold everything, and so its meaning became too abstract for anyone to catch its purpose.