= Solution
Perform the <elementary row operations> \b[from the bottom upwards]:
$$
R_4\gets R_4-tR_3,\qquad R_3\gets R_3-tR_2,\qquad R_2\gets R_2-tR_1.
$$
This order matters because each subtraction uses the predecessor's original row. For example,
$$
(t^3,t^2,t,1)-t(t^2,t,1,t)=(0,0,0,1-t^2).
$$
The other two differences are $(0,0,1-t^2,t-t^3)$ and $(0,1-t^2,t-t^3,t^2-t^4)$. Thus the required <upper triangular matrix> is
$$
\boxed{U=\begin{pmatrix}
1&t&t^2&t^3\\
0&1-t^2&t(1-t^2)&t^2(1-t^2)\\
0&0&1-t^2&t(1-t^2)\\
0&0&0&1-t^2
\end{pmatrix}.}
$$
These <elementary row operations> use no division, so the reduction remains valid for all <real numbers> $t$, including $t=\pm1$.
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