FINDING INVERSE OF 3X3 MATRIX EXAMPLES

Let A be a square matrix of order n. If there exists a square matrix B of order n such that

AB  =   BA  =  I_n

then the matrix B is called an inverse of A.

Note :

Let A be square matrix of order n. Then, A⁻¹ exists if and only if A is non-singular.

Formula to find inverse of a matrix

Finding Inverse of 2 x 2 Matrix

Example 1 :

Find the inverse (if it exists) of the following:

Since |A|  =  2 ≠ 0, it is non singular matrix. A^-1 exists.

Finding the Inverse of a 3x3 Matrix Examples

Example 2 :

Solution :

In order to find inverse of a matrix, first we have to find |A|.

|A|  =  5(25 - 1) - 1(5 - 1) + 1(1 - 5)

  =  5(24 ) - 1(4) + 1(-4)

  =  120 - 4 - 4

  =  112

Since |A|  =  112 ≠ 0, it is non singular matrix. A^-1 exists.

Example 3 :

Solution :

In order to find inverse of a matrix, first we have to find |A|.

|A|  =  2(8 - 7) - 3(6 - 3) + 1(21 - 12)

  =  2(1) - 3(3) + 1(9)

  =  2 - 9 + 9

  =  2

Since |A|  =  2 ≠ 0, it is non singular matrix. A^-1 exists.

Example 4 :

Solution :

Let A  =  F(α)

A^-1 =  (1/|A|) adj A

|A|  =  cos α [cos α - 0] - 0[0 - 0] + sin α[0 + sin α]

  =  cos²α + sin²α

|A|  =  1

Hence proved.

Example 5 :

Solution :

  =  A² - 3A - 7I₂

Finding the value of A² :

Finding the value of 3A :

Finding the value of 7I₂ :

A² - 3A - 7I₂ 

Hence proved.

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