No, that is not the only way to represent "i". Another obvious choice would be but there are many others.

I presume that you are representing the real number, a, by the matrix so that -1 is represented as . So if [tex]\begin{pmatrix} a & b \\ c & d \end{pmatrix}[tex], where a, b, c, and d are real numbers, represents the "square root of negative one", we must have

and so must have , , , and .

From ab+ bd= b(a+ d)= 0, we must have either b= 0 or a+ d= 0. If b= 0 then becomes which is impossible so d= -a.

From ac+ cd= c(a+ d)= 0 we get the same thing. The other two equations then become so that .

Thesimplestthing to do is to take a= d= 0 and either b= 1, c= -1 or b=-1, c= 1 but there many other choices.