Inverse to 3x3 matrix
Name: michael j stevenson
Date: Around 1995
How do you find the inverse of a 3x3 matrix? My Algebra II book does a
wonderful job of explaining matrices in general and working with the
determinant and inverse of 2x2 matrices, but it is very vague about 3x3.
Although the book supplies us with the inverse if it is needed, I would like
to know how to do it for myself.
The reason that your book is so vague about the inverse of a three by three
matrix is that they wanted to avoid the several pages of explanation needed
to show how to find the inverse of a larger matrix. There are two commonly
used methods. One uses determinants (you need to be able to find determi-
nants of square matrices of size bigger than 2 by 2) and the other uses row
operations. The second method is computationally the best. It may be time
for some reading on your part. Try to find an easy reading linear algebra
There are, of course, formulas for calculating the inverses of matrices
of any size. In order to understand these formulas, you need to create some
examples. So create a 3x3 matrix using algebra symbols for the entries. It
will look like this:
a b c
d e f
g h i
Now write out the inverse matrix in terms of the 9 unknowns: rstuvwxyz.
The product of the two matrices must give the matrix:
1 0 0
0 1 0
0 0 1
so you end up with 9 equations in 9 unknowns. So the formulas just give you
a concise way of writing the solutions of these 9 equations.
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Update: June 2012