# 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.

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

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

A best free mathematics education website for students, teachers and researchers.

###### Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

###### Maths Problems

Learn how to solve the math problems in different methods with understandable steps and worksheets on every concept for your practice.

Learn solutions

###### Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.