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$$$\left[\begin{array}{ccc}0 & \frac{1}{2} & 0\\\frac{\sqrt{2}}{4} & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{4}\\\frac{\sqrt{2}}{4} & 0 & \frac{\sqrt{2}}{4}\end{array}\right]\cdot \left[\begin{array}{c}\frac{\sqrt{6}}{6}\\\frac{\sqrt{6}}{3}\\\frac{\sqrt{6}}{6}\end{array}\right]$$$

The calculator will multiply the $$$3$$$x$$$3$$$ matrix $$$\left[\begin{array}{ccc}0 & \frac{1}{2} & 0\\\frac{\sqrt{2}}{4} & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{4}\\\frac{\sqrt{2}}{4} & 0 & \frac{\sqrt{2}}{4}\end{array}\right]$$$ by the $$$3$$$x$$$1$$$ matrix $$$\left[\begin{array}{c}\frac{\sqrt{6}}{6}\\\frac{\sqrt{6}}{3}\\\frac{\sqrt{6}}{6}\end{array}\right]$$$, with steps shown.

Related calculator: Matrix Calculator

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Your Input

Calculate $$$\left[\begin{array}{ccc}0 & \frac{1}{2} & 0\\\frac{\sqrt{2}}{4} & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{4}\\\frac{\sqrt{2}}{4} & 0 & \frac{\sqrt{2}}{4}\end{array}\right]\cdot \left[\begin{array}{c}\frac{\sqrt{6}}{6}\\\frac{\sqrt{6}}{3}\\\frac{\sqrt{6}}{6}\end{array}\right].$$$

Solution

$$$\left[\begin{array}{ccc}{\color{GoldenRod}0} & {\color{DarkMagenta}\frac{1}{2}} & {\color{Red}0}\\{\color{Peru}\frac{\sqrt{2}}{4}} & {\color{Crimson}\frac{\sqrt{2}}{2}} & {\color{SaddleBrown}\frac{\sqrt{2}}{4}}\\{\color{Fuchsia}\frac{\sqrt{2}}{4}} & {\color{DarkBlue}0} & {\color{BlueViolet}\frac{\sqrt{2}}{4}}\end{array}\right]\cdot \left[\begin{array}{c}{\color{Violet}\frac{\sqrt{6}}{6}}\\{\color{Purple}\frac{\sqrt{6}}{3}}\\{\color{Brown}\frac{\sqrt{6}}{6}}\end{array}\right] = \left[\begin{array}{c}{\color{GoldenRod}\left(0\right)}\cdot {\color{Violet}\left(\frac{\sqrt{6}}{6}\right)} + {\color{DarkMagenta}\left(\frac{1}{2}\right)}\cdot {\color{Purple}\left(\frac{\sqrt{6}}{3}\right)} + {\color{Red}\left(0\right)}\cdot {\color{Brown}\left(\frac{\sqrt{6}}{6}\right)}\\{\color{Peru}\left(\frac{\sqrt{2}}{4}\right)}\cdot {\color{Violet}\left(\frac{\sqrt{6}}{6}\right)} + {\color{Crimson}\left(\frac{\sqrt{2}}{2}\right)}\cdot {\color{Purple}\left(\frac{\sqrt{6}}{3}\right)} + {\color{SaddleBrown}\left(\frac{\sqrt{2}}{4}\right)}\cdot {\color{Brown}\left(\frac{\sqrt{6}}{6}\right)}\\{\color{Fuchsia}\left(\frac{\sqrt{2}}{4}\right)}\cdot {\color{Violet}\left(\frac{\sqrt{6}}{6}\right)} + {\color{DarkBlue}\left(0\right)}\cdot {\color{Purple}\left(\frac{\sqrt{6}}{3}\right)} + {\color{BlueViolet}\left(\frac{\sqrt{2}}{4}\right)}\cdot {\color{Brown}\left(\frac{\sqrt{6}}{6}\right)}\end{array}\right] = \left[\begin{array}{c}\frac{\sqrt{6}}{6}\\\frac{\sqrt{3}}{2}\\\frac{\sqrt{3}}{6}\end{array}\right]$$$

Answer

$$$\left[\begin{array}{ccc}0 & \frac{1}{2} & 0\\\frac{\sqrt{2}}{4} & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{4}\\\frac{\sqrt{2}}{4} & 0 & \frac{\sqrt{2}}{4}\end{array}\right]\cdot \left[\begin{array}{c}\frac{\sqrt{6}}{6}\\\frac{\sqrt{6}}{3}\\\frac{\sqrt{6}}{6}\end{array}\right] = \left[\begin{array}{c}\frac{\sqrt{6}}{6}\\\frac{\sqrt{3}}{2}\\\frac{\sqrt{3}}{6}\end{array}\right]\approx \left[\begin{array}{c}0.408248290463863\\0.866025403784439\\0.288675134594813\end{array}\right]$$$A