The question as asked in the title is incorrect. It is not important for a matrix to be square. Matrices are defined to solve specific problems, and the specifi

The question as asked in the title is incorrect. It is not important for a matrix to be square. Matrices are defined to solve specific problems, and the specific problem is what determines the geometry, not some goal of having square matrices. However the person actually asked why it is important for matrices to be square for most of the theorems they are learning. The actual answers to this are interesting, but the presence of this question likely implies that they are being taught abstract theory first prior to building much intuition. I was taught this way as well in my advanced pure math course. It was all super abstract until I was studying for the final exam and then had this eureka moment where suddenly everything made sense (a matrix is just a numerical way of describing a linear transform! And computing eigenvectors is like factoring!). Sadly, this happened again the next term where we derived SVD - except this time it never made sense to me until a later on course where we needed to use SVD for some application.