Chủ Nhật, 20 tháng 12, 2015

Inverse of a Matrix using Minors, Cofactors and Adjugate 3x3

We can calculate the Inverse of a Matrix by:
  • Step 1: calculating the Matrix of Minors,
  • Step 2: then turn that into the Matrix of Cofactors,
  • Step 3: then the Adjugate, and
  • Step 4: multiply that by 1/Determinant.
But it is best explained by working through an example!
Example: find the Inverse of A:
It needs 4 steps. It is all simple arithmetic but there is a lot of it, so try not to make a mistake!
Step 1: Matrix of Minors
The first step is to create a "Matrix of Minors". This step has the most calculations:
Determinant
For a 2×2 matrix (2 rows and 2 columns) the determinant is easy: ad-bc
Think of a cross:
  • Blue means positive (+ad),
  • Red means negative (-bc)
A Matrix
(It gets harder for a 3×3 matrix, etc)

The Calculations

Here are the first two, and last two, calculations of the "Matrix of Minors" (notice how I ignore the values in the current row and columns, and calculate the determinant using the remaining values):
And here is the calculation for the whole matrix:

Step 2: Matrix of Cofactors

This is easy! Just apply a "checkerboard" of minuses to the "Matrix of Minors". In other words, we need to change the sign of alternate cells, like this:

Step 3: Adjugate (also called Adjoint)

Now "Transpose" all elements of the previous matrix... in other words swap their positions over the diagonal (the diagonal stays the same):

Step 4: Multiply by 1/Determinant

Now find the determinant of the original matrix. This isn't too hard, because we already calculated the determinants of the smaller parts when we did "Matrix of Minors".
A Matrix
So: multiply the top row elements by their matching "minor" determinants:
Determinant = 3×2 - 0×2 + 2×2 = 10
And now multiply the Adjugate by 1/Determinant:
And we are done!

Không có nhận xét nào:

Đăng nhận xét