$$\begin{array}{ll} \text{minimize} & R_1^2+R_2^2\\ \text{subject to} & RT=I\end{array}$$ where $R\in\mathbb{R^{2\times2}}$, $T\in\mathbb{R^{2\times2}}$, $R=\begin{bmatrix}R_1 & R_2\\R_3 &R_4\end{bmatrix}$, $T=\begin{bmatrix}T_1 & T_2\\T_3 &T_4\end{bmatrix}$.
My attempt: I know that if $RT=I$ then $\min R^{}R^*=(T^*T^{})^{-1}$. Then $R_1^2+R_2^2 \geq (T^*T^{})^{-1}_{1\times 1}$, where $(T^*T^{})^{-1}_{1\times1}$ is the first diagonal element of $(T^*T^{})^{-1}$. Can we get tighter bound?
We can make $R_1^2 + R_2^2$ arbitrarily close to $0$. To show this, consider the identity, for any $x \not= 0$, $$\begin{bmatrix}0 & 1/x \\ 1/x &0 \end{bmatrix}\begin{bmatrix}0 & x \\ x & 0\end{bmatrix} = I$$