Determinants — Definition, Minors and Cofactors

Every square matrix $A$ has associated with it a single real number called its determinant, written $|A|$ or $\det(A)$. Determinants let us test whether a system of linear equations has a unique solution, and are the key ingredient in finding a matrix's inverse (Section 6).

Determinant of a $2\times 2$ Matrix

Determinant of a $3\times 3$ Matrix

For $A=\begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{pmatrix}$ expanding along the first row:

Expansion can be done along any row or column (with the sign pattern below) and always gives the same value — a useful cross-check.

Minors

The minor $M_{ij}$ of the element $a_{ij}$ is the determinant of the smaller matrix left after deleting the row and column that $a_{ij}$ sits in.

Cofactors

The cofactor $C_{ij}$ attaches a sign to the minor: $C_{ij} = (-1)^{i+j} M_{ij}$ The sign pattern for a $3\times 3$ matrix is

$\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}$ …