Question

Which of the following matrix is not orthogonal?

(a) \(\begin{bmatrix}0.33 & 0.67 & -0.67\\ -0.67 & 0.67 & 0.33\\ 0.67 & 0.33 & 0.67\end{bmatrix} \)

(b) \(\begin{bmatrix}cosx &sinx \\-sinx & cosx\end{bmatrix} \)

(c) \(\begin{bmatrix}0.33 & -0.67 & 0.67\\0.67 & 0.67 & 0.33\\ -0.67 & 0.33 & 0.67\end{bmatrix} \)

(d) \(\begin{bmatrix}cosx & sinx \\-sinx & -cosx \end{bmatrix} \)

This question was addressed to me in quiz.

The origin of the question is Transformation (Reduction) of Quadratic Form to Canonical Form in division Eigen Values and Eigen Vectors of Engineering Mathematics

Answer 1

The correct answer is (a) \(\begin{bmatrix}0.33 & 0.67 & -0.67\\ -0.67 & 0.67 & 0.33\\ 0.67 & 0.33 & 0.67\end{bmatrix} \)

The explanation is: Out of the given options, \(\begin{bmatrix}0.33 & 0.67 & -0.67\\ -0.67 & 0.67 & 0.33\\ 0.67 & 0.33 & 0.67\end{bmatrix} \) satisfies the condition for orthogonality, i.e. AA^T = I

\(\begin{bmatrix}0.33 & 0.67 & -0.67\\ -0.67 & 0.67 & 0.33\\ 0.67 & 0.33 & 0.67\end{bmatrix}

\begin{bmatrix}0.33 & -0.67 & 0.67 \\ 0.67 & 0.67 & 0.33\\-0.67 & 0.33 & 0.67\end{bmatrix}= \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} \)

Since \(\begin{bmatrix}0.33 & -0.67 & 0.67\\ 0.67 & 0.67 & 0.33\\ -0.67 & 0.33 & 0.67\end{bmatrix} \)is the transpose of A, it is also orthogonal.

Coming to \(\begin{bmatrix}cosx & sinx\\ -sin x &cosx\end{bmatrix}, \)

\(\begin{bmatrix}cosx & sinx\\-sinx & cosx\end{bmatrix} \begin{bmatrix}cosx & -sinx \\ sinx & cosx\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} \)

The remaining option \(\begin{bmatrix}cosx & sinx\\ -sinx & -cosx\end{bmatrix}, \)which does not satisfy the condition for orthogonality.