= Solution
{c}
Assume $A\ne0$, 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 $c_1,\ldots,c_n$ such that
$$
E_{22}=E_{21}E_{12}
=\sum_{k=1}^n c_kE_{12}^{\,k}E_{21}.
$$
But $E_{12}^2=0$ and $E_{12}E_{21}=E_{11}$, so the right side is $c_1E_{11}$. Its $(2,2)$ entry is zero, whereas the $(2,2)$ entry of $E_{22}$ is $1_A\ne0$. 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.
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