Thornton & Marion (5th Edition), Chapter 01, Exercise Problem 05

Thornton & Marion, Classical Dynamics, Fifth Edition

Chapter 1. Matrices, Vectors, and Vector Calculus

Problem 05. The determinant of the transformation matrix

The problem asks you to
show that |\bold{\lambda}|^2 = 1
This problem assumes
the transformation matrix \bold{\lambda} to be a two-dimensional orthogonal matrix.
You should know about
1) Calculation of a determinant
2) Properties of the orthogonal transformation matrix
Solution
Since the transformation matrix \bold{\lambda} is

\bold{\lambda} = \left( \begin{array}{cc}  \lambda_{11} & \lambda_{12} \\ \lambda_{21} & \lambda_{22} \end{array} \right)
then determinant of this matrix is
|\lambda| = \lambda_{11}\lambda_{22} - \lambda_{12}\lambda_{21}
So
|\lambda|^2 = (\lambda_{11}\lambda_{22} - \lambda_{12}\lambda_{21})^2 = \lambda_{11}^2\lambda_{22}^2 + \lambda_{12}^2\lambda_{21}^2 - 2\lambda_{11}\lambda_{22}\lambda_{12}\lambda_{21}
From Equation (1.13), which is the orthogonality condition,
\sum_j = \lambda_{ij}\lambda_{kj} = \delta_{ik}
|\lambda|^2 = \lambda_{11}^2\lambda_{22}^2 = 1
Therefore,
|\bold{\lambda}|^2 = 1

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