Cofactors of a 3×3 Matrix
Fact-checked:
Definition
The minor of a three by three matrix with a sign is called the cofactor of an entry in a square of the order three.
Introduction
Let’s consider a $3 \times 3$ matrix, denoted by $A$.
$A$ $\,=\,$ $\begin{bmatrix} e_{11} & e_{12} & e_{13} \\ e_{21} & e_{22} & e_{23} \\ e_{31} & e_{32} & e_{33} \\ \end{bmatrix}$
The cofactor of an element in a matrix of order $3$ is a product of the following factors.
- The negative one raised to the power of sum of “the number of the row” and “the number of the column” of the respective entry.
- The minor of the corresponding element.
$C_A$ $\,=\,$ $\begin{bmatrix} (-1)^{1+1} \times M_{11} & (-1)^{1+2} \times M_{12} & (-1)^{1+3} \times M_{13} \\ (-1)^{2+1} \times M_{21} & (-1)^{2+2} \times M_{22} & (-1)^{2+3} \times M_{23} \\ (-1)^{3+1} \times M_{31} & (-1)^{3+2} \times M_{32} & (-1)^{3+3} \times M_{33} \\ \end{bmatrix}$
$\therefore\,\,\,$ $C_A$ $\,=\,$ $\begin{bmatrix} M_{11} & -M_{12} & M_{13} \\ -M_{21} & M_{22} & -M_{23} \\ M_{31} & -M_{32} & M_{33} \\ \end{bmatrix}$
