A fact about complexity of algorithms for computing the product of matrices was brought up to me that was very interesting I was not aware of. I still am not sure what the optimal bound is on the minimum number of scalar multiplications to compute a product of $n \times n$ matrices (maybe it is a simple result in the $2 \times 2$ case?) but I was told there is at least a basic proof for the following case. After playing around for a while I was not sure what the best route to take was so I thought I would just ask what is the easiest way to show the following result:
How do you prove that if $A,B$ are two $2 \times 2$ matrices then the product $AB$ can be computed using only 7 scalar multiplications?