The basis vectors can be represented by matrices, this algebra was worked out independently by Pauli for his work on quantum mechanics, we can define the following equivalents:
The scalar would be the identity matrix.
The bivector can be calculated by multiplying the matrices:
So the complete geometric multiplication table is:
a*b 
b.e 
b.e1 
b.e2 
b.e12 
a.e 




a.e1 




a.e2 




a.e12 




which is equivilant to the table derived here.
Any multivector can be represented by a single matrix representing a linear sum of these basis, for instance,

= 3* 

+ 4* 

= 3e_{1}+ 4e_{2} 
Further Reading
Other uses of Pauli Matrix:
Related Concepts: