For me, the most amusing feature of LA is how quickly results and tools escalate from trivial and obvious to surprising and hard. LA is often introduced as slick notation to express systems of linear equations. Gaussian elimination and matrix inverses (when they exist) are natural evolutions of this, but not what I consider surprising.
Matrix multiplication is an elementary next step, but it is not at all clear why one would multiply matrices. My first real mathematical surprise (I was in jr high at the time) is a result of matrix multiplication: there is a matrix A of real values that satisfies A @ A = -I. (The matrix is [[0,1],[-1,0]]). I was aware of complex numbers, but until that moment, I didn't know you could get "minus identity" without first artificially introducing the idea of "square root of -1". But this shows one can construct "square root of -1" without introducing anything like that.
When one gets to determinants, characteristic polynomials, eigenstructure, fixed subspaces, projection operators, SVD, orthogonal transformations, and so-on, the work really begins. That determinants are multiplicative is really wonderful, but not at all obvious. SVD in full generality was only shown in 1936 and modern algorithms emerged in the 1950's to compute it [1].
Finally, check out the LDU decomposition of the Walsh matrix [2].
[1] https://en.wikipedia.org/wiki/Singular_value_decomposition#H...