For any integral domain $k$ , the group of $n\times n$ invertible matrices with coefficients in $k[t]$ is the amalgamated free product of invertible matrices over $k$ and invertible upper triangular matrices over $k[t]$ , amalgamated over the upper triangular matrices of $k$ . More compactly $$\GL{n}{(k[t])}\cong\GL {n}{(k)} *_{B(k)}B(k[t]).$$