# Minors of entries in a matrix

The determinant of the sub-matrix by leaving the row and the column of an entry is called the minor of that element.

## Introduction

There are some entries in each matrix and each element has one minor. The minor of every element is calculated from the concept of determinant of a matrix.

For finding the minor of an entry in a matrix, the elements in the row of that entry and also the entries in the column of the same entry are neglected. Thus, it forms a square matrix with the remaining entries and the determinant of that matrix is called the minor of the entry.

$A$ $\,=\,$ $\begin{bmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n}\\ a_{21} & a_{22} & a_{23} & \cdots & a_{2n}\\ a_{31} & a_{32} & a_{33} & \cdots & a_{3n}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & a_{n3} & \cdots & a_{nn} \end{bmatrix}$

Let’s understand the concept of the minor of an entry by considering the entry $a_{11}$. The minor of element $a_{11}$ in the matrix $A$ is denoted by $M_{11}$ in matrix algebra.

The entry $a_{11}$ is an entry in the first row and also an entry in the first column in the matrix of the order $n$. Hence, leave the elements in the first row and also leave the entries in the first column. It forms a sub-square matrix. Find the determinant of that matrix and it is called the minor of the entry $a_{11}$ in the matrix $A$.

$M_{11}$ $\,=\,$ $\begin{vmatrix} a_{22} & a_{23} & \cdots & a_{2n}\\ a_{32} & a_{33} & \cdots & a_{3n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{n2} & a_{n3} & \cdots & a_{nn} \end{vmatrix}$

Thus, the minor of any entry in a square matrix can be calculated by following this procedure.

### Examples

Let’s learn the concept of the minor of an entry from the following understandable example cases.

#### Minors of a 2×2 matrix

Learn how to find the minor of every entry in a two by two square matrix.

#### Minors of a 3×3 matrix

Learn how to evaluate the minor of each element in a three by three matrix.

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