How would I find the 3×3 matrix A that satisfies the following? $$A\begin{pmatrix}1\\0\\0\end{pmatrix}=\begin{pmatrix}2\\-5\\ 1\end{pmatrix}$$
$$A\begin{pmatrix}0\\1\\0\end{pmatrix}=\begin{pmatrix}-1\\2\\-5\end{pmatrix}$$
$$A\begin{pmatrix}0\\0\\1\end{pmatrix}=\begin{pmatrix}-4\\-1\\5\end{pmatrix}$$
I find this problem particularly confusing as you can't find the inverse of any of these 3×1 matrices.

Since $A$ is being multiplied by the three standard basis vectors of $\Bbb R^3$, just concatenate the result column vectors together:
$$A=\begin{bmatrix}2&-1&-4\\-5&2&-1\\1&-5&5\end{bmatrix}$$
Verify that this satisfies the three given equations.
