Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Digital systems, linear algebra, introduction to formal proof / 6 / b / Solution

A unit vector along the line is $d=(3,4{)}^{\top }/5$. Reflection preserves the orthogonal projection $d{d}^{\top }u$ onto that line and negates its perpendicular component. Therefore $Au=d{d}^{\top }u-(u-d{d}^{\top }u)$ andAs 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.