Complexity Explorer Santa Fe Institute

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Vector and Matrix Algebra

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2.3 Essential Types of Matrices » Quiz #11 Solution

Question 1:  As stated in the video, a matrix Q is orthogonal if QT = Q-1

Here, A^T=\begin{bmatrix} \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{bmatrix},

and

(A^T)(A)=\begin{bmatrix} \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{bmatrix} \begin{bmatrix} \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{bmatrix} = \begin{bmatrix} 1 & 0 & 0___ \\ 0 & 1 & 0 __ \\ 0 & 0 & 1 \end{bmatrix}.  So we can conclude that AT = A-1 and thus A is orthogonal.