Minor of Matrices

In a square matrix, each element possesses its own minor. The minor is defined as a value obtained from the determinant of a square matrix by deleting out a row and a column corresponding to the element of a matrix.

Given a square matrix A, by minor of an element  , we mean the value of the determinant obtained by deleting the  row and  column of A matrix. It is denoted by . In order to find the minor of the square matrix, we have to erase out a row & a column one by one at the time & calculate their determinant, until all the minors are computed. The following are the steps to calculate minor from a matrix:

1. Hide  row and  column one by one from given matrix, where i refer to m and j refers to n that is the total number of rows and columns in matrices.
2. Evaluate the value of the determinant of the matrix made after hiding a row and a column from Step 1.

Minor of 3×3 Matrix

Consider the 3*3 matrix  We had to hide the first row and column in order to find the minors of matrices.

= ei – hf

= di – fg

= dh – eg

= bi – ch

= ai – cg

= ah – bg

= bf – ce

= af – cd

= ae – bd

Example: Consider the 3*3 matrix 

Solution: We first calculate minor of element 2. Since it is (1,1) element of A, we delete first row and first column, so that determinant of remaining array is  = (4*-2) – (2*-1) = -8+2= -6 = 

Since -1 is (1,2) element, we delete first row and second column. The determinant of remaining array  = 0*-2-(2*1) = -2 = 

The minor of 3 is  = 0-4 = -4 =  

The minor of 0 is  = (-1)(-2)-(3)(-1) = 2+3 = 5 = 

The minor of 4 is  = (2)(-2)-(3)(1) = -4-3 = -7

The minor of 2 in (2,3) place in  = (2)(-1) – (1)(1) = -2+1 = -1 = 

The minor of 1 is  = (-1)(2) – (3)(4) = -2-12 = -14 = 

The minor of (-1) is  = (4)-0 = 4 =

The minor of (-2) is  = (2)(4)-0 = 8  = 

Minor of 2×2 Matrix

For a 2*2 matrix, calculation of minors is very simple. Let us consider a 2 x 2 matrix . We had to hide the first row and column to find the minors of matrices.

= d

= c

= b

= a

Example: Consider the matrix . For finding minor of 2 we delete first row and first column.

Solution: . So that remaining array is |7| = 7 = 

Similarly, minors of 6, -4 and 7 will be -4,6,2 respectively.

Exercise

1. Find the minor of the matrix .
2. Find the minor of matrix .
3. Find the minor of matrix.
4. Find the minor of matrix .
5. Find the minor of the matrix .