How do I write an arbitrary 2×2 matrix as a linear combination of the three Pauli Matrices and the 2×2 unit matrix?

Any example for the same might help ?

**Answer**

A slow construction would go…

(abcd)=a(1000)+b(0100)+c(0010)+d(0001)

(1000)=12(1001)+12(100−1)=1212+12σ3

(0100)= ...

⟹(abcd)=a212+a2σ3+ ... (other combintations of the four matrices)

**Attribution***Source : Link , Question Author : chakra , Answer Author : Nikolaj-K*