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 and in such cases the Gauss elimination approach is prefered.
To obtain using the determinant approach the steps in the following keypoint are followed:
Key Point 10
Matrix Inverse the Determinant Method
Given a square matrix :
- Find . If then does not exist. If we can proceed to find the inverse matrix, as follows.
- Replace each element of by its cofactor (see Section 7.3).
- Transpose the result to form the adjoint matrix , denoted by adj
- Then calculate adj .
Task!
Find the inverse of . This will require five stages.
-
First find
:
-
Now replace each element of
by its minor:
-
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
-
to obtain the matrix of cofactors:
(d) Then transpose the result to obtain the adjoint matrix:
Transposing, adj
-
Finally obtain
:
as before using Gauss elimination.
Exercises
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Find the inverses of the following matrices
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Use the determinant method and also the Gauss elimination method to find the inverse of the following matrices