Cofactors of a 3×3 Matrix

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.

1. 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.
2. The minor of the corresponding element.

$C_A$ $\,=\,$ $\begin{bmatrix} (-1)^{1+1} \times M_{11} & (-1)^{1+2} \times M_{12} & (-1)^{1+3} \times M_{13} \\ (-1)^{2+1} \times M_{21} & (-1)^{2+2} \times M_{22} & (-1)^{2+3} \times M_{23} \\ (-1)^{3+1} \times M_{31} & (-1)^{3+2} \times M_{32} & (-1)^{3+3} \times M_{33} \\ \end{bmatrix}$

$\therefore\,\,\,$ $C_A$ $\,=\,$ $\begin{bmatrix} M_{11} & -M_{12} & M_{13} \\ -M_{21} & M_{22} & -M_{23} \\ M_{31} & -M_{32} & M_{33} \\ \end{bmatrix}$

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