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.