Department of Energy Argonne National Laboratory Office of Science NEWTON's Homepage NEWTON's Homepage
NEWTON, Ask A Scientist!
NEWTON Home Page NEWTON Teachers Visit Our Archives Ask A Question How To Ask A Question Question of the Week Our Expert Scientists Volunteer at NEWTON! Frequently Asked Questions Referencing NEWTON About NEWTON About Ask A Scientist Education At Argonne Inverse to 3x3 matrix
Name: michael j stevenson
Status: N/A
Age: N/A
Location: N/A
Country: N/A
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 text.


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.


Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (, or at Argonne's Educational Programs

Educational Programs
Building 360
9700 S. Cass Ave.
Argonne, Illinois
60439-4845, USA
Update: June 2012
Weclome To Newton

Argonne National Laboratory