Question

Find the values of \( x, y, z \) if the matrix \( A \) satisfies the equation \( A^T A = I \), where

\[ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} \]

Hint:

For an orthogonal matrix \( A \), the condition \( A^T A = I \) means each column vector of \( A \) must be orthonormal.

Answer 1

Step 1: Compute \( A^T \) (transpose of \( A \)).

\[ A^T = \begin{bmatrix} 0 & x & x \\ 2y & y & -y \\ -z & -z & z \end{bmatrix} \]

Step 2: Compute \( A^T A \).

\[ A^T A = \begin{bmatrix} x^2 + x^2 & x y + x y & x z + x z \\ y x + y x & 4y^2 + y^2 + y^2 & -2y z - y z + y z \\ z x + z x & -2y z - y z + y z & z^2 + z^2 + z^2 \end{bmatrix} \]

Step 3: Solve for \( x, y, z \) using \( A^T A = I \). Comparing with identity matrix:

\[ \begin{aligned} & 2x^2 = 1 \Rightarrow x = \pm \frac{1}{\sqrt{2}}, \\ & 6y^2 = 1 \Rightarrow y = \pm \frac{1}{\sqrt{6}}, \\ & 3z^2 = 1 \Rightarrow z = \pm \frac{1}{\sqrt{3}}. \end{aligned} \]