Science:Math Exam Resources/Courses/MATH152/April 2017/Question A 21/Solution 1

Following the hint, we know that $A$ is an orthogonal matrix. That is, the two column vectors are orthogonal to each other and have norms of $1$.

The orthogonal condition implies that $a/2-b/2=0$

and the norm condition implies $a^{2}+(1/2)^{2}=1,b^{2}+(1/2)^{2}=1.$

By solving these equations, we get a solution:

$\color {blue}{a=b=\pm {\frac {\sqrt {3}}{2}}}$.