Let us take a scenario,

Imagine a school assembly where children are asked to stand in a equal number of lines as per neat arrangement which explains discipline and coordination. 

This will be an equivalent example of the term matrix.

1. MATRIX: The arrangement of numbers in rows and columns in a rectangular array. Mathematicians Gauss, Jordan, Cayley, and Hamilton have developed the theory of matrices which has been used in investigating solutions of systems of linear equations.

2. Here we will learn about 4 important methods:
   1. Matrix inversion method, 
   2. Crammers rule,
   3. Gaussian elimination method, and 
   4. Rank method.

Let’s recollect the basics of this chapter to experiment with these concepts.

Let us consider a matrix 

and calculate its inverse.

SOLUTION:

STEPS:

1. Arrange the matrix based on the first row, first row second column, and so on.
2. Find the inverse of each element in the matrix.
3. interpretation.

Let’s make it manually

• let us see if the determinant of the matrix is zero to make sure we can inverse it.
• Since it is a 3✕3 matrix let’s write the elements as follows and find the determinants as follows:

The formula for the determinant is given as follows 

Let us apply the formula to check the determinant is not  0 so that it is an inverse matrix.

Since the determinant is not zero let us proceed with the next step.

Let’s find minor for the given matrix.

• MINOR OF A11=0.3

NEGLECT THE FIRST ROW AND FIRST COLUMN:

• MINOR OF A12

NEGLECT THE FIRST ROW AND SECOND COLUMN:

• MINOR OF A13

NEGLECT THE FIRST ROW AND THIRD COLUMN

• MINOR OF A21

NEGLECT THE SECOND ROW AND FIRST COLUMN

• MINOR OF A22

NEGLECT THE SECOND ROW AND SECOND COLUMN

• MINOR OF A23

NEGLECT THE SECOND ROW AND THIRD COLUMN

• MINOR OF A31

NEGLECT THE THIRD ROW AND FIRST COLUMN

• MINOR OF A32

NEGLECT THE THIRD ROW AND SECOND COLUMN

• MINOR OF A33

NEGLECT THE THIRD ROW AND THIRD COLUMN

THUS WE GET

MINOR OF A11=0.18

MINOR OF A12=0.00

MINOR OF A13=-0.09

MINOR OF A21=0.07

MINOR OF A22=0.10

MINOR OF A23=-0.01

MINOR OF A31=-0.01

MINOR OF A32=0.05

MINOR OF A33=0.13

• Lets take cofactors from the minors
• Now lets take transpose 
• It is nothing but taking adjoint matrix