Solution
= Solution
A <unit vector> along the line is $d=(3,4)^\top/5$. Reflection preserves the <orthogonal projection> $dd^\top u$ onto that line and negates its perpendicular component. Therefore $Au=dd^\top u-(u-dd^\top u)$ and
$$
\boxed{A=2dd^\top-I=\frac1{25}\begin{pmatrix}-7&24\\24&7\end{pmatrix}.}
$$
As checks, $A(3,4)^\top=(3,4)^\top$, $A(-4,3)^\top=-(-4,3)^\top$, and $A^2=I$. Its <determinant> is minus one, as expected for a planar <reflection matrix>.