# equivalence between matrix multiplication and matrix inversion

I am a bit confused with this wikipedia article, hoping someone can clarify it. Looking at the Strassen algorithm page it is clear that this is an algorithm for reducing multiplication operations.

Why is a matrix multiplication algorithm touted as a matrix inversion algorithm? last time i checked it was totally odd to apply Gauss-Jordan elimination to multiply matrices

thanks

• A small tweak of the Strassen algorithm can be used to invert a matrix; see the bottom of the mathworld page. The wikipedia page is comparing the Gauss-Jordan method of matrix inversion with the Strassen method of matrix inversion; and the Strassen method of matrix multiplication with the usual, direct, method of matrix multiplication (there is no indication whatsoever that they are doing "Gauss-Jordan" for multiplication). Jun 22, 2011 at 21:02
• thanks, it says that the formula applies only to $2 \times 2$ matrices? or it just refers to the $2 \times 2$ block partitioning into submatrices? Jun 22, 2011 at 21:10
• I don't know the algorithm myself well enough; I programmed Strassen's algorithm for multiplication once, but that was over 20 years ago; but google search gives, for example, the code Mathematica uses. I think you just recursively go down to the 2x2 case, where the problem can be solved directly, and then use Strassen's method to "lift" it to a full matrix. Jun 22, 2011 at 21:13
• that makes sense; that would be the answer to my question. If you paste in answer format i will accept it Jun 22, 2011 at 21:21

Gauss Jordan algorithm is effecting manipulations on the lines of the matrix to invert. Each operation is equivalent to multiply your matrix on the left side by a matrix of the form $I_n +\lambda E_{ij}$ where $E_{ij}$ is an elementary matrix: $E_{ij}=(e^{ij}_{kl})_{1\leq k,l\leq n}$ where $e^{ij}_{kl}=1$ if $k=i$ and $l=j$, and $0$ otherwise.
So each time you are making such an operation, you are actually multiplying your matrix by some other matrix, so all your line operations are actually equivalent to multiply your matrix by a series of matrix of the form $I_n +\lambda E_{ij}$.