Solution: The matrix $\mathbfM$ is constructed by placing the images of the standard basis vectors as its columns. Thus, $\mathbfM = \beginpmatrix 2 & 3 \\ -1 & 4 \endpmatrix$. Verifying, $\mathbfM \beginpmatrix 1 \\ 0 \endpmatrix = \beginpmatrix 2 \\ -1 \endpmatrix$ and $\mathbfM \beginpmatrix 0 \\ 1 \endpmatrix = \beginpmatrix 3 \\ 4 \endpmatrix$, confirming correctness. $\boxed\beginpmatrix 2 & 3 \\ -1 & 4 \endpmatrix$ - Appfinity Technologies