Let \(\alpha\) be a solution of \({x}^{2}+x+1=0\), and for some \(a\) and \(b\) in \(\mathrm{ℝ},\) \(\left[\begin{matrix…

Let \(\alpha\) be a solution of \({x}^{2}+x+1=0\), and for some \(a\) and \(b\) in \(\mathrm{ℝ},\) \(\left[\begin{matrix}4 & \mathrm{a} & \mathrm{b}\end{matrix}\right]\left[\begin{matrix}1 & 16 & 13 \\ -1 & -1 & 2 \\ -2 & -14 & -8\end{matrix}\right]=\left[\begin{matrix}0 & 0 & 0\end{matrix}\right].\) If \(\frac{4}{{\alpha }^{4}}+\frac{\mathrm{m}}{{\alpha }^{\mathrm{a}}}+\frac{\mathrm{n}}{{\alpha }^{\mathrm{b}}}=3,\) then \(m+n\) is equal to

[JEE Main 2025, 8 Apr (Shift 1)]