Determinants

To every square matrix $A=[a_{ij}]$ of order $n$, we associate a single number called the determinant of $A$, written $|A|$ (also $\det(A)$ or $\det A$ or $\Delta$). Determinants are defined only for square matrices — there is no determinant of a $2\times 3$ matrix, say.

It is important not to confuse the two ideas: a matrix is a representation — an array of numbers — while a determinant is a single value computed from a square matrix. If $A=[a_{ij}]_{n\times n}$ then $|A|=\begin{vmatrix}a_{11}&\cdots&a_{1n}\\ \vdots&\ddots&\vdots\\ a_{n1}&\cdots&a_{nn}\end{vmatrix}$ — the same array of numbers, but written between vertical …