Explanation
\begin{aligned}
& \mathrm{A}^{2}=\mathrm{I}\Longrightarrow|\mathrm{A}|^{2}=\mathrm{I}\Rightarrow|\mathrm{A}|=\pm1=b \\
& \mathrm{Let~A}=
\begin{bmatrix}
\alpha & \beta \\
\gamma & \delta
\end{bmatrix} \\
& \Rightarrow\mathrm{A}^{2}=
\begin{bmatrix}
\alpha & \beta \\
\gamma & \delta
\end{bmatrix}
\begin{bmatrix}
\alpha & \beta \\
\gamma & \delta
\end{bmatrix}=\mathrm{I} \\
& \Rightarrow\mathrm{A}^{2}=
\begin{bmatrix}
\alpha^{2}+\beta\gamma & \alpha\beta+\beta\delta \\
\alpha\gamma+\gamma\delta & \gamma\beta+\delta^{2}
\end{bmatrix}
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
\end{aligned}
\begin{aligned}
& \alpha\beta+\beta\delta=0\Rightarrow(\alpha+\delta)\beta=0\Rightarrow\alpha+\delta=0=a \\
& (\alpha+\delta)\gamma=0\Longrightarrow\beta\gamma+\delta^{2}=0 \\
& \mathrm{Now}3a^{2}+4b^{2}=3(0)^{2}+4(1)=4
\end{aligned}
