Inverting 3x3 Matrices

1
Inverting 3x3 Matrices
Consider the matrix
First we need to find the cofactors. These are
found by crossing out the row/column of an entry
and calculating the 2x2 determinant created, and
then finding the correct sign. For instance A1
the cofactor of a1 is
2
Inverting 3x3 Matrices
The signs used are in a chess board pattern
Thus A2 the cofactor of a2 is
Etc.
3
Determinant of a 3x3 Matrix
The determinant can now be defined in terms of
the cofactors as det a1A1b1B1c1C1 or det
a1A1a2A2a3A3 det a2A2b2B2c2C2 or det
b1B1b2B2b3B3 det a3A3b3B3c3C3 or det
c1C1c2C2c3C3
4
Alien Cofactors
Note that if you calculate the product of entries
with a different row/column then instead of
getting the determinant you get zero
a1A2b1B2c1C2 0 or c1A1c2A2c3A3 0 This
result is called the property of alien cofactors
(Dr Who beware!) Why is this the case? Basically
because you have made two rows or two columns
the same, and thus the determinant has to be zero
as there will no longer be a unique solution.
5
Finding the inverse

• Find the matrix of the cofactors
• Transpose the matrix, by swapping the rows and
  columns
• Divide by the determinant

6
Finding the inverse
The method on the previous slide is very prone to
errors, so check your result. Do this by
evaluating the product
It should be
This is also one way to calculate the
determinant.