How do I decipher the notation in Sakurai's Modern QM problem?

[inha]

I need help with deciphering notation from the second exercise of Sakurai's Modern QM's first chapter. Here's how it's presented in the book:

Suppose a 2x2 matrix X (not neccessarily Hermitian, nor unitary) is written as
X=a_0+\sigma \cdot a,
where a_0 and a_k (k=1,2,3) are numbers.

a. How are a_0 and a_k related to tr(X) and tr(\sigma_k X )
b. Obtain a_0 and a_k in terms of the matrix elements X_{ij}

Now I have no idea what the matrix X is supposed to look like. Nor can I even figure out how a 2x2 matrix could be written like that. I remember seeing someone ask something about the same exercise here but I couldn't find that thread via search. I can't really present any work here since I don't know what the matrix is supposed to look like but could someone help me get started with this anyway?

 

[qbert]

The main point here is that the Pauli matrices form a
complete set. The notation is shorthand for:
X = a_0 I + a_1 \sigma_1 + a_2 \sigma_2 + a_3 \sigma 3
or
X = a_0 \left( \begin{array}{cc} 1 &amp; 0 \\ 0 &amp; 1 \end{array} \right)<br /> + a_1 \left( \begin{array}{cc} 0 &amp; 1 \\ 1 &amp; 0 \end{array} \right)<br /> + a_2 \left( \begin{array}{cc} 0 &amp; -i \\ i &amp; 0 \end{array} \right)<br /> + a_3 \left( \begin{array}{cc} 1 &amp; 0 \\ 0 &amp; -1 \end{array} \right)<br />

Now to solve the problem you'll want to use the following facts
Tr(A+B) = Tr(A) + Tr(B)
Tr(sigma_i) = 0
sigma_i . sigma_i = I
sigma_1. sigma_2 = i sigma_3 (and even permutations).

 

[inha]

Thanks a lot! I didn't realize that the sigmas were supposed to be the Pauli matrices and that got me confused.