Why Can't Scalars and Matrices Be Added in Pauli Matrix Calculations?

[frerk]

Homework Statement

Hey :-)
I just need some help for a short calculation.
I have to show, that
$$(\sigma \cdot a)(\sigma \cdot b) = (a \cdot b) + i \sigma \cdot (a \times b)$$

The Attempt at a Solution

I am quiet sure, that my mistake is on the right side, so I will show you my calculation for this one:
$$a_xb_x + a_yb_y+a_zb_z + i\sigma_x (a_yb_z - a_3b_2) + i\sigma_y (a_zb_x-a_xb_z) + i\sigma_z (a_xb_y -a_yb_x)$$

The last 3 terms are a 2x2 matrix and the first 3 terms are just a scalar...
So i can`t add them.

would be happy fora small hint what is wrong :-)
Thank you

 

[blue_leaf77]

There is actually an identity matrix to be multiplied wit $a\cdot b$.

 

[frerk]

There is actually an identity matrix to be multiplied wit $a\cdot b$.

hey. thank you for your answer.
Yes, right. that brings to to the result I want.
Is there a rule, why I have to multiply the result of the dot product with the idendity matrix?
Because the other terms include a Pauli Matrix and the result
of the dot produkt must adapt to that structure?

 

[blue_leaf77]

hey. thank you for your answer.
Yes, right. that brings to to the result I want.
Is there a rule, why I have to multiply the result of the dot product with the idendity matrix?
Because the other terms include a Pauli Matrix and the result
of the dot produkt must adapt to that structure?

Of course it can be proven using the more fundamental properties of Pauli matrices, especially their commutation and anti-commutation. An easy prove can be found here.