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:

- 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. - 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

- Find the minor of the matrix .
- Find the minor of matrix .
- Find the minor of matrix.
- Find the minor of matrix .
- Find the minor of the matrix .

Anurag says

Error in “Minor of 3*3 Matrix” Example M11. Should be “ei-hf” not “ef-hi”

Maths Tutor says

Fixed.

sean says

second the first comment, major error that could negatively affect students’ understanding of minors. please change or delete this page to prevent further damage.