Español | Português
  1. Home
  2. Calculators
  3. Calculators: Linear Algebra
  4. Linear Algebra Calculator

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

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

$$$\times$$$
$$$\times$$$

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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

$$$\left[\begin{array}{ccc}{\color{Green}0} & {\color{DarkCyan}\frac{1}{2}} & {\color{Chartreuse}0}\\{\color{Fuchsia}\frac{\sqrt{2}}{4}} & {\color{Crimson}\frac{\sqrt{2}}{2}} & {\color{DarkBlue}\frac{\sqrt{2}}{4}}\\{\color{BlueViolet}\frac{\sqrt{2}}{4}} & {\color{Brown}0} & {\color{GoldenRod}\frac{\sqrt{2}}{4}}\end{array}\right]\cdot \left[\begin{array}{c}{\color{Chartreuse}\frac{\sqrt{6}}{6}}\\{\color{DarkCyan}\frac{\sqrt{6}}{3}}\\{\color{Brown}\frac{\sqrt{6}}{6}}\end{array}\right] = \left[\begin{array}{c}{\color{Green}\left(0\right)}\cdot {\color{Chartreuse}\left(\frac{\sqrt{6}}{6}\right)} + {\color{DarkCyan}\left(\frac{1}{2}\right)}\cdot {\color{DarkCyan}\left(\frac{\sqrt{6}}{3}\right)} + {\color{Chartreuse}\left(0\right)}\cdot {\color{Brown}\left(\frac{\sqrt{6}}{6}\right)}\\{\color{Fuchsia}\left(\frac{\sqrt{2}}{4}\right)}\cdot {\color{Chartreuse}\left(\frac{\sqrt{6}}{6}\right)} + {\color{Crimson}\left(\frac{\sqrt{2}}{2}\right)}\cdot {\color{DarkCyan}\left(\frac{\sqrt{6}}{3}\right)} + {\color{DarkBlue}\left(\frac{\sqrt{2}}{4}\right)}\cdot {\color{Brown}\left(\frac{\sqrt{6}}{6}\right)}\\{\color{BlueViolet}\left(\frac{\sqrt{2}}{4}\right)}\cdot {\color{Chartreuse}\left(\frac{\sqrt{6}}{6}\right)} + {\color{Brown}\left(0\right)}\cdot {\color{DarkCyan}\left(\frac{\sqrt{6}}{3}\right)} + {\color{GoldenRod}\left(\frac{\sqrt{2}}{4}\right)}\cdot {\color{Brown}\left(\frac{\sqrt{6}}{6}\right)}\end{array}\right] = \left[\begin{array}{c}\frac{\sqrt{6}}{6}\\\frac{\sqrt{3}}{2}\\\frac{\sqrt{3}}{6}\end{array}\right]$$$

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