4. Determinants

To doubt everything, or, to believe everything, are two equally convenient solutions; both dispense with the necessity of reflection.

– Henri Poincaré (1854 - 1912)

The purpose of this section is to define and study the basic properties of the most important invariant associated to any square matrix $A\in M_n(ℝ)$, called the determinant of $A$, which is written $detA$. In Section 3, we saw that $ad-bc\ne 0$ if and only if the $2\times 2$ matrix was invertible. For $n\times n$ matrices, we will see that $detA\ne 0$ if and only if $A$ is invertible. In this section, we develop the algorithm, based on row and column operations on matrices, to compute $detA$ of square matrices of any size. We start with the case of $2\times 2$ matrices, and use that case to build the definition of the determinant for larger matrices.

Throughout this section, all matrices are square.