Resolved: "Identify orthogonal matrices Invert orthogonal matrice [1323232313−2323−2313]"

Joni Kenny

Answered question

2020-12-09

Identify orthogonal matrices Invert orthogonal matrice

[\begin{matrix} \frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\ \frac{2}{3} & \frac{1}{3} & - \frac{2}{3} \\ \frac{2}{3} & - \frac{2}{3} & \frac{1}{3} \end{matrix}]

Answer & Explanation

berggansS

Skilled2020-12-10Added 91 answers

Step 1
To identify the orthogonal matrix

[\begin{matrix} \frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\ \frac{2}{3} & \frac{1}{3} & - \frac{2}{3} \\ \frac{2}{3} & - \frac{2}{3} & \frac{1}{3} \end{matrix}]

Step 2
Note that a matrix is orthogonal if it satisfies:

A A^{T} = I

A = [\begin{matrix} \frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\ \frac{2}{3} & \frac{1}{3} & - \frac{2}{3} \\ \frac{2}{3} & - \frac{2}{3} & \frac{1}{3} \end{matrix}] \Rightarrow A^{T} = [\begin{matrix} \frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\ \frac{2}{3} & \frac{1}{3} & - \frac{2}{3} \\ \frac{2}{3} & - \frac{2}{3} & \frac{1}{3} \end{matrix}]

A A^{T} = [\begin{matrix} \frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\ \frac{2}{3} & \frac{1}{3} & - \frac{2}{3} \\ \frac{2}{3} & - \frac{2}{3} & \frac{1}{3} \end{matrix}] [\begin{matrix} \frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\ \frac{2}{3} & \frac{1}{3} & - \frac{2}{3} \\ \frac{2}{3} & - \frac{2}{3} & \frac{1}{3} \end{matrix}]

= [\begin{matrix} \frac{1}{9} + \frac{4}{9} + \frac{4}{9} & \frac{2}{9} + \frac{2}{9} - \frac{4}{9} & \frac{2}{9} - \frac{4}{9} + \frac{2}{9} \\ \frac{2}{9} + \frac{2}{9} - \frac{4}{9} & \frac{4}{9} + \frac{1}{9} + \frac{4}{9} & \frac{4}{9} - \frac{2}{9} - \frac{2}{9} \\ \frac{2}{9} - \frac{4}{9} + \frac{2}{9} & \frac{4}{9} - \frac{2}{9} - \frac{2}{9} & \frac{4}{9} + \frac{1}{9} + \frac{4}{9} \end{matrix}]

= [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = I

Step 3
Thus matrix is orthogonal.
Now note that the inverse of orthogonal matrix can be find as:

A A^{T} = I

\Rightarrow A^{- 1} = A^{T}

\Rightarrow A^{- 1} = [\begin{matrix} \frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\ \frac{2}{3} & \frac{1}{3} & - \frac{2}{3} \\ \frac{2}{3} & - \frac{2}{3} & \frac{1}{3} \end{matrix}]

Jeffrey Jordon

Expert2022-01-27Added 2575 answers

Answer is given below (on video)

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