# 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$$|\bold{\lambda}|^2 = 1$
##### This problem assumes
the transformation matrix $\bold{\lambda}$$\bold{\lambda}$ to be a two-dimensional orthogonal matrix.
1) Calculation of a determinant
2) Properties of the orthogonal transformation matrix
##### Solution
Since the transformation matrix $\bold{\lambda}$$\bold{\lambda}$ is

$\bold{\lambda} = \left( \begin{array}{cc} \lambda_{11} & \lambda_{12} \\ \lambda_{21} & \lambda_{22} \end{array} \right)$$\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}$$|\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}$$|\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}$$\sum_j = \lambda_{ij}\lambda_{kj} = \delta_{ik}$

$|\lambda|^2 = \lambda_{11}^2\lambda_{22}^2 = 1$$|\lambda|^2 = \lambda_{11}^2\lambda_{22}^2 = 1$

Therefore,

$|\bold{\lambda}|^2 = 1$$|\bold{\lambda}|^2 = 1$