$$$\left[\begin{array}{ccc}\frac{\sqrt{6}}{6} & \frac{\sqrt{6}}{6} & \frac{\sqrt{6}}{3}\\\frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & - \frac{\sqrt{3}}{3}\\\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & 0\end{array}\right]\cdot \left[\begin{array}{ccc}1 & 3 & 5\\1 & 3 & 1\\2 & -1 & 7\end{array}\right]$$$

Calculate $$$\left[\begin{array}{ccc}\frac{\sqrt{6}}{6} & \frac{\sqrt{6}}{6} & \frac{\sqrt{6}}{3}\\\frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & - \frac{\sqrt{3}}{3}\\\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & 0\end{array}\right]\cdot \left[\begin{array}{ccc}1 & 3 & 5\\1 & 3 & 1\\2 & -1 & 7\end{array}\right].$$$

$$$\left[\begin{array}{ccc}{\color{Peru}\frac{\sqrt{6}}{6}} & {\color{BlueViolet}\frac{\sqrt{6}}{6}} & {\color{OrangeRed}\frac{\sqrt{6}}{3}}\\{\color{Chocolate}\frac{\sqrt{3}}{3}} & {\color{Red}\frac{\sqrt{3}}{3}} & {\color{Brown}- \frac{\sqrt{3}}{3}}\\{\color{Magenta}\frac{\sqrt{2}}{2}} & {\color{DarkBlue}- \frac{\sqrt{2}}{2}} & {\color{GoldenRod}0}\end{array}\right]\cdot \left[\begin{array}{ccc}{\color{BlueViolet}1} & {\color{OrangeRed}3} & {\color{GoldenRod}5}\\{\color{DarkBlue}1} & {\color{DarkCyan}3} & {\color{Fuchsia}1}\\{\color{Brown}2} & {\color{Magenta}-1} & {\color{DarkMagenta}7}\end{array}\right] = \left[\begin{array}{ccc}{\color{Peru}\left(\frac{\sqrt{6}}{6}\right)}\cdot {\color{BlueViolet}\left(1\right)} + {\color{BlueViolet}\left(\frac{\sqrt{6}}{6}\right)}\cdot {\color{DarkBlue}\left(1\right)} + {\color{OrangeRed}\left(\frac{\sqrt{6}}{3}\right)}\cdot {\color{Brown}\left(2\right)} & {\color{Peru}\left(\frac{\sqrt{6}}{6}\right)}\cdot {\color{OrangeRed}\left(3\right)} + {\color{BlueViolet}\left(\frac{\sqrt{6}}{6}\right)}\cdot {\color{DarkCyan}\left(3\right)} + {\color{OrangeRed}\left(\frac{\sqrt{6}}{3}\right)}\cdot {\color{Magenta}\left(-1\right)} & {\color{Peru}\left(\frac{\sqrt{6}}{6}\right)}\cdot {\color{GoldenRod}\left(5\right)} + {\color{BlueViolet}\left(\frac{\sqrt{6}}{6}\right)}\cdot {\color{Fuchsia}\left(1\right)} + {\color{OrangeRed}\left(\frac{\sqrt{6}}{3}\right)}\cdot {\color{DarkMagenta}\left(7\right)}\\{\color{Chocolate}\left(\frac{\sqrt{3}}{3}\right)}\cdot {\color{BlueViolet}\left(1\right)} + {\color{Red}\left(\frac{\sqrt{3}}{3}\right)}\cdot {\color{DarkBlue}\left(1\right)} + {\color{Brown}\left(- \frac{\sqrt{3}}{3}\right)}\cdot {\color{Brown}\left(2\right)} & {\color{Chocolate}\left(\frac{\sqrt{3}}{3}\right)}\cdot {\color{OrangeRed}\left(3\right)} + {\color{Red}\left(\frac{\sqrt{3}}{3}\right)}\cdot {\color{DarkCyan}\left(3\right)} + {\color{Brown}\left(- \frac{\sqrt{3}}{3}\right)}\cdot {\color{Magenta}\left(-1\right)} & {\color{Chocolate}\left(\frac{\sqrt{3}}{3}\right)}\cdot {\color{GoldenRod}\left(5\right)} + {\color{Red}\left(\frac{\sqrt{3}}{3}\right)}\cdot {\color{Fuchsia}\left(1\right)} + {\color{Brown}\left(- \frac{\sqrt{3}}{3}\right)}\cdot {\color{DarkMagenta}\left(7\right)}\\{\color{Magenta}\left(\frac{\sqrt{2}}{2}\right)}\cdot {\color{BlueViolet}\left(1\right)} + {\color{DarkBlue}\left(- \frac{\sqrt{2}}{2}\right)}\cdot {\color{DarkBlue}\left(1\right)} + {\color{GoldenRod}\left(0\right)}\cdot {\color{Brown}\left(2\right)} & {\color{Magenta}\left(\frac{\sqrt{2}}{2}\right)}\cdot {\color{OrangeRed}\left(3\right)} + {\color{DarkBlue}\left(- \frac{\sqrt{2}}{2}\right)}\cdot {\color{DarkCyan}\left(3\right)} + {\color{GoldenRod}\left(0\right)}\cdot {\color{Magenta}\left(-1\right)} & {\color{Magenta}\left(\frac{\sqrt{2}}{2}\right)}\cdot {\color{GoldenRod}\left(5\right)} + {\color{DarkBlue}\left(- \frac{\sqrt{2}}{2}\right)}\cdot {\color{Fuchsia}\left(1\right)} + {\color{GoldenRod}\left(0\right)}\cdot {\color{DarkMagenta}\left(7\right)}\end{array}\right] = \left[\begin{array}{ccc}\sqrt{6} & \frac{2 \sqrt{6}}{3} & \frac{10 \sqrt{6}}{3}\\0 & \frac{7 \sqrt{3}}{3} & - \frac{\sqrt{3}}{3}\\0 & 0 & 2 \sqrt{2}\end{array}\right]$$$

$$$\left[\begin{array}{ccc}\frac{\sqrt{6}}{6} & \frac{\sqrt{6}}{6} & \frac{\sqrt{6}}{3}\\\frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & - \frac{\sqrt{3}}{3}\\\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & 0\end{array}\right]\cdot \left[\begin{array}{ccc}1 & 3 & 5\\1 & 3 & 1\\2 & -1 & 7\end{array}\right] = \left[\begin{array}{ccc}\sqrt{6} & \frac{2 \sqrt{6}}{3} & \frac{10 \sqrt{6}}{3}\\0 & \frac{7 \sqrt{3}}{3} & - \frac{\sqrt{3}}{3}\\0 & 0 & 2 \sqrt{2}\end{array}\right]\approx \left[\begin{array}{ccc}2.449489742783178 & 1.632993161855452 & 8.16496580927726\\0 & 4.04145188432738 & -0.577350269189626\\0 & 0 & 2.82842712474619\end{array}\right]$$$A