Step 1: Basicity and Lanthanoid Contraction: - As we move across the lanthanide series, atomic size decreases due to lanthanoid contraction.
- \( La(OH)_3 \) has a larger ionic radius, making it more basic than \( Lu(OH)_3 \).
Step 2: Explanation: - Larger ionic size in \( La^{3+} \) leads to weaker bond strength in \( La(OH)_3 \), making hydroxide ions (\( OH^- \)) more available.
- \( Lu^{3+} \) has a smaller radius, leading to a stronger attraction between \( Lu^{3+} \) and \( OH^- \), reducing basicity.
Find the values of \( x, y, z \) if the matrix \( A \) satisfies the equation \( A^T A = I \), where
\[ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} \]