Matrices

On the surface a matrix is just numbers in a grid. What makes it worth studying is what it does: an m×n matrix encodes a linear transformation taking n-dimensional vectors to m-dimensional ones. Multiplying a matrix by a vector applies the transformation; the grid is the bookkeeping.

That view explains the multiplication rule, which otherwise looks arbitrary. The entry in row i, column j of AB is the dot product of A's row i with B's column j — not the product of corresponding entries, which is what addition does and what most people expect. The rule is what it is because AB has to mean "do B, then do A", and composition of transformations forces exactly this arithmetic. It also explains the shape requirement: A's columns must match B's rows, or the composition is meaningless.

Order therefore matters. Rotating then reflecting is not the same as reflecting then rotating, so AB ≠ BA in general — an inequality that has no analogue in ordinary arithmetic and that catches almost everyone at least once. Division has no analogue either. Instead there is the inverse matrix A⁻¹ undoing A, and it exists only when the transformation is reversible. A matrix that flattens space onto a line has destroyed information no inverse can recover, and its determinant is zero — which is exactly what the determinant is for.