4 The inverse of a 3 × 3 matrix - determinant method

This method which employs determinants, is of importance from a theoretical perspective. The numerical computations involved are too heavy for matrices of higher order than 3 × 3 and in such cases the Gauss elimination approach is prefered.

To obtain A 1 using the determinant approach the steps in the following keypoint are followed:

Key Point 10

Matrix Inverse the Determinant Method

Given a square matrix A :

  • Find | A | . If | A | = 0 then A 1 does not exist. If | A | 0 we can proceed to find the inverse matrix, as follows.
  • Replace each element of A by its cofactor (see Section 7.3).
  • Transpose the result to form the adjoint matrix , denoted by adj ( A )
  • Then calculate A 1 = 1 A adj ( A ) .
Task!

Find the inverse of A = 0 1 1 2 3 1 1 2 1 . This will require five stages.

  1. First find A :

    A = 0 × 5 + 1 × ( 1 ) + 1 × 7 = 6

  2. Now replace each element of A by its minor:

    3 1 2 1 2 1 1 1 2 3 1 2 1 1 2 1 0 1 1 1 0 1 1 2 1 1 3 1 0 1 2 1 0 1 2 3 = 5 1 7 1 1 1 4 2 2

  3. Now attach the signs from the array

    + + + + +

    (so that where a + sign is met no action is taken and where a sign is met the sign is change

  4. to obtain the matrix of cofactors:

    5 1 7 1 1 1 4 2 2

    (d) Then transpose the result to obtain the adjoint matrix:

    Transposing, adj ( A ) = 5 1 4 1 1 2 7 1 2

  5. Finally obtain A 1 :

    A 1 = 1 det ( A ) adj ( A ) = 1 6 5 1 4 1 1 2 7 1 2 as before using Gauss elimination.

Exercises
  1. Find the inverses of the following matrices
    1. 1 2 3 4
    2. 1 0 0 4
    3. 1 1 1 1
  2. Use the determinant method and also the Gauss elimination method to find the inverse of the following matrices
    1. A = 2 1 0 1 0 0 4 1 2
    2. B = 1 1 1 0 1 1 0 0 1
    1. 1 2 4 2 3 1
    2. 1 0 0 1 4
    3. 1 2 1 1 1 1
    1. A 1 = 1 2 0 2 1 2 4 2 0 0 1 T = 1 2 0 2 0 2 4 0 1 2 1
    2. B 1 = 1 0 0 1 1 0 0 1 1 T = 1 1 0 0 1 1 0 0 1