Cofactor of an Element of a Determinant

The cofactor of an element refines its minor with a sign that depends on position. For the element in row $i$, column $j$, the cofactor is denoted $A_{ij}$ (or $C_{ij}$) and defined as

where $M_{ij}$ is the minor of that element. The sign alternates in a checkerboard pattern across the matrix: $+, -, +, -, \dots$ starting from $a_{11}$.

Cofactors are what let you evaluate a determinant of any order: to find $\det A$ for a $3 \times 3$ matrix, pick any one row (or column), multiply each of its elements by its own cofactor, and add the results — expanding along the first row, for instance, gives $\det A = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}$. Remarkably, expanding along any of the three rows or any of the three columns always gives the same value. …