# Thornton & Marion, Classical Dynamics of Particles and Systems, 5th Edition

## Chapter 1. Matrices, Vectors, and Vector Calculus

### Problem 05. The determinant of the transformation matrix

#### The problem asks you to

• show that $latex |\\bold{\\lambda}|^2 = 1$

#### This problem assumes

• The transformation matrix $latex \\bold{\\lambda}$ to be a two-dimensional orthogonal matrix.

1. Calculation of a determinant
2. Properties of the orthogonal transformation matrix

#### Solution

Since the transformation matrix $latex \\bold{\\lambda}$ is

$latex \\bold{\\lambda} = \\left( \\begin{array}{cc} \\lambda_{11} & \\lambda_{12} \\\\ \\lambda_{21} & \\lambda_{22} \\end{array} \\right)$
then determinant of this matrix is

$latex |\\lambda| = \\lambda_{11}\\lambda_{22} – \\lambda_{12}\\lambda_{21}$

So

$latex |\\lambda|^2 = (\\lambda_{11}\\lambda_{22} – \\lambda_{12}\\lambda_{21})^2 = \\lambda_{11}^2\\lambda_{22}^2 + \\lambda_{12}^2\\lambda_{21}^2 – 2\\lambda_{11}\\lambda_{22}\\lambda_{12}\\lambda_{21}$

From Equation (1.13), which is the orthogonality condition,

$latex \\sum_j = \\lambda_{ij}\\lambda_{kj} = \\delta_{ik}$
$latex |\\lambda|^2 = \\lambda_{11}^2\\lambda_{22}^2 = 1$

Therefore,

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