 Solution

Assume $A\ne 0$, as is implicit in the claimed conclusion. Suppose that a surjective ring homomorphism $f:R\to M_2(A)$ exists. Choose $r_1,r_2\in R$ mapping to the matrix units $E_{12}$ and $E_{21}$. Applying $f$ to the assumed relation gives integers $c_1,\dots ,c_n$ such thatBut $E_{12}^2=0$ and $E_{12}{E}_{21}=E_{11}$, so the right side is $c_1{E}_{11}$. Its $(2,2)$ entry is zero, whereas the $(2,2)$ entry of $E_{22}$ is $1_A\ne 0$. This contradiction proves that no such surjective ring homomorphism exists.

If the convention for rings allows $A$ to be the zero ring, that case must be excluded from the assertion: $M_2(0)=0$, and the zero ring homomorphism is surjective. This is a convention issue in the printed question.