LinArg assignment 10

exercise

10.2 Orthogonal 2 × 2 matrices and rotation matrices

a)

$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

b)

$A$ is orthogonal $⟹$ $A^{\top }A=I$ $⟹\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ $⟹\{\begin{array}{l}{a}^2+c^2=1\\ ab+cd=0\\ b^2+d^2=1\end{array}$ $⟹(a^2+c^2)(b^2+d^2)-(ab+cd{)}^2=1\cdot 1-0\cdot 0=1$ $⟹a^2{b}^2+a^2{d}^2+b^2{c}^2+c^2{d}^2-a^2{b}^2-c^2{d}^2-2abcd=1$ $⟹a^2{d}^2+b^2{c}^2-2abcd=1$ $⟹(ad{)}^2+(bc{)}^2-2(ad)(bc)=1$ $⟹(ad-bc{)}^2=1$ $⟹ad-bc=\pm 1$ $⟹\det (A)=\pm 1$ $⟹$ $|\det (A)|=1$

c)

$A=\left[\begin{array}{cc}1& 2\\ 1& 3\end{array}\right]$ not orthogonal, but $|\det (A)|=|1\cdot 3-1\cdot 2|=1$