Complexity Explorer Santa Few Institute

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 Q^T= 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 A^T= A^-1 and thus A is orthogonal.