I generally try to understand mathematical concepts as thoroughly as I can, and there's one thing I've never quite been able to understand intuitively.

Take the system

x = 2a + 3b + c
y = 3a +4b + 2c
z = a + b + 2c

and solve for a, b and c

a = -6x + 5y - 2z
b = 4x - 3y + z
c = x - y + z

Now the two matricies representing these systems are the inverse of one another, and thus if we multiply them we get the identity matrix. This is the part I don't get; why for example the constants 2*(-6) + 3*4 + 1*1 neccesarily equal 1 etc.

I can show that it is so in using vectors and thinking of it that way, but I don't intuitively "see" it. Any enlightenment is appreciated!