Mathematics - Matrices Question with Solution | TestHub

$\begin{array}{rl}& A=\left[\begin{array}{lll}(\sqrt{2}{)}^2& (\sqrt{2}{)}^3& (\sqrt{2}{)}^4\\ (\sqrt{2}{)}^3& (\sqrt{2}{)}^4& (\sqrt{2}{)}^5\\ (\sqrt{2}{)}^4& (\sqrt{2}{)}^5& (\sqrt{2}{)}^6\end{array}\right]\\ & A=\left[\begin{array}{ccc}2& 2\sqrt{2}& 4\\ 2\sqrt{2}& 4& 4\sqrt{2}\\ 4& 4\sqrt{2}& 8\end{array}\right]\\ & A^2=2^2\left[\begin{array}{ccc}1& \sqrt{2}& 2\\ \sqrt{2}& 2& 2\sqrt{2}\\ 2& 2\sqrt{2}& 4\end{array}\right]\left[\begin{array}{ccc}1& \sqrt{2}& 2\\ \sqrt{2}& 2& 2\sqrt{2}\\ 2& 2\sqrt{2}& 4\end{array}\right]\end{array}$

$=4\left[\begin{array}{ccc}-& -& -\\ -& -& -\\ (2+4+8)& (2\sqrt{2}+4\sqrt{2}+8\sqrt{2})& (4+8+16)\end{array}\right]$




Sum of elements of $3^{\text{nd }}$ row $=4(14+14\sqrt{2}+28)$
$\begin{array}{rl}& =4(42+14\sqrt{2})\\ & =168+56\sqrt{2}\\ & \alpha +\beta \sqrt{2}\\ \therefore & \alpha +\beta =168+56=224\end{array}$