square matrix

A square matrix is one whose grid is literally square: it has the same number of rows as columns. A 2-by-2, a 3-by-3, a 100-by-100 — all square. A 2-by-3 is not.

We call an n-by-n square matrix a matrix of order n. The entries a_11, a_22, a_33, … running from the top-left corner down to the bottom-right form the main diagonal, a feature unique to square matrices.

Squareness unlocks a whole toolkit. Only square matrices have a determinant, only square matrices can possibly have an inverse, and the identity matrix is square by definition. Whenever you hear about invertibility, determinants, or eigenvalues, a square matrix is hiding in the background.