Orthogonal matrices

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   Let $a = 5$, $b = 12$, and $c = 13$, so $a^{2}+b^{2}=c^{2}$, and let $R = \begin{pmatrix} 5/13 & -12/13 \\ 12/13 & 5/13 \end{pmatrix}$. $R$ is orthogonal, so $R^{-1} = R^{T}$. Find the row $2$, column $1$ entry of $R^{-1}$.

2. Warm-up

   Let $a = 21$, $b = 20$, and $c = 29$, so $a^{2}+b^{2}=c^{2}$, and let $R = \begin{pmatrix} 21/29 & -20/29 \\ 20/29 & 21/29 \end{pmatrix}$. $R$ is orthogonal, so $R^{-1} = R^{T}$. Find the row $2$, column $1$ entry of $R^{-1}$.

3. Warm-up

   Let $a = 3$, $b = 4$, and $c = 5$, so $a^{2}+b^{2}=c^{2}$, and let $R = \begin{pmatrix} 3/5 & -4/5 \\ 4/5 & 3/5 \end{pmatrix}$. $R$ is orthogonal, so $R^{-1} = R^{T}$. Find the row $2$, column $1$ entry of $R^{-1}$.

Problems shown under their original terms: Generated (template, solver-computed).