You don't need matrices really, only use the Extended Euclidean algorithm. Here is a layout for it. I continued one line further, in order to obtain the quotient of the two given numbers by their gcd:
Thus you have a solution: $\;(x,y)=(-22\,303,1090)$. The theory tells us all other solutions have the form:
$$(x,y)=\Bigl(-22\,303+k44\,872,1090-k2193),\quad k\in\mathbf Z.$$
Not only the gcd is a linear combination of the two given numbers, but all intermediate remainders (easy induction). Si if $r_i+1$ denotes the remainder and $q_i$ the quotient in the $i$-th division (initialisation for the remainders: $r_0=89\,744$, $r_1=4386$), we have
and of course the same relation is valid for the coefficients $u_i,v_i$, by linearity.