Minors and Cofactors

Minors

For a square matrix $A$ of order $3$, the minor $M_{ij}$ of the element $a_{ij}$ is the determinant of the $2\times2$ matrix left after deleting the row and the column that $a_{ij}$ sits in -- that is, row $i$ and column $j$. Every entry of a $3\times3$ matrix has its own minor, so a $3\times3$ matrix has nine minors in total, $M_{11}$ through $M_{33}$. (For a $2\times2$ matrix, the "minor" of an entry is simply the single remaining entry after deleting that entry's row and column.)

Cofactors

The cofactor $C_{ij}$ of the element $a_{ij}$ attaches a sign to the minor $M_{ij}$:

The sign $(-1)^{i+j}$ follows a fixed checkerboard pattern across the matrix, starting with $+$ in the top-left corner:

So $C_{11}=+M_{11}$, $C_{12}=-M_{12}$, $C_{13}=+M_{13}$, $C_{21}=-M_{21}$, $C_{22}=+M_{22}$, and so on: whenever $i+j$ is even the cofactor equals the minor unchanged, and whenever $i+j$ is odd the cofactor is the negative of the minor.

Expansion Along Any Row or Column, Restated with Cofactors

Section 1's expansion formula can now be written compactly as

and, as promised in Section 1, expanding along any row $i$ or any column $j$ gives the identical value:

This is why choosing a row or column with the most zeros (Section 1's tip) is always safe: whichever row or column is chosen, every entry's cofactor is multiplied in, and a zero entry contributes nothing to the sum regardless of its cofactor's value -- so the zero row/column is simply skipped over in the arithmetic. …