Notation
On this page we number the matrix elements with two index numbers the first represents the column, the second the row. The index numbers start at 0 as follows:
m00
m10
m20
m01
m11
m21
m02
m12
m22
Other notation conventions are explained on this page.
Description
The invese of a 3x3 matrix is:
[M]
-1 = 1/det[M] *
m11*m22 - m12*m21
m02*m21 - m01*m22
m01*m12 - m02*m11
m12*m20 - m10*m22
m00*m22 - m02*m20
m02*m10 - m00*m12
m10*m21 - m11*m20
m01*m20 - m00*m21
m00*m11 - m01*m10
where:
det = m00*m11*m22 + m01*m12*m20 + m02*m10*m21 - m00*m12*m21 - m01*m10*m22 - m02*m11*m20
Speed Optimisation
The above program is valid for a general 3×3 matrix which will work in all circumstances but when the matrix is being used to represent a rotation (as described on this page) then the matrix carries a lot of redundant information. So if we want to speed up the code on this page then, for this case only, we can take advantage of this redundant information. To invert a pure rotation then we just take the transpose of the 3x3 part of the matrix.
Calculator
The following calculator allows you to calculate the inverse for a 3x3 matrix. Enter the values into the matrix and then press "calc inverse " to display the result:
Code
public void inverse() {
double det = m00*m11*m22 + m01*m12*m20 + m02*m10*m21 - m00*m12*m21 - m01*m10*m22 - m02*m11*m20;
m00 = (m11*m22 - m12*m21)/det;
m01 = (m02*m21 - m01*m22)/det;
m02 = (m01*m12 - m02*m11)/det;
m10 = (m12*m20 - m10*m22)/det;
m11 = (m00*m22 - m02*m20)/det;
m12 = (m02*m10 - m00*m12)/det;
m20 = (m10*m21 - m11*m20)/det;
m21 = (m01*m20 - m00*m21)/det;
m22 = (m00*m11 - m01*m10)/det;
}
Further Information
Book Shop - Further reading.
Where I can, I have put links to Amazon for books that are relevant to
the subject, click on the appropriate country flag to get more details
of the book or to buy it from them.
cover
Developing Games in Java
This site may have errors. Don't use for critical systems.