Applications of Matrices Multiplication to Determinant and Rotations formulas in $\setR^n$

TL;DR

This paper provides a straightforward proof of the determinant's multiplicative property, a constructive rotation formula, and classifies invariant subspaces of 4D equiangular rotations, making these topics accessible in elementary linear algebra.

Contribution

It introduces a simple proof of the determinant property, a constructive rotation matrix formula, and classifies invariant subspaces in 4D rotations, enhancing understanding of linear algebra concepts.

Findings

Simple proof of determinant multiplicative property

Constructive formula for rotation matrices

Classification of invariant subspaces in 4D rotations

Abstract

This note deals with two topics of linear algebra. We give a simple and short proof of the multiplicative property of the determinant and provide a constructive formula for rotations. The derivation of the rotation matrix relies on simple matrix calculations and thus can be presented in an elementary linear algebra course. We also classify all invariant subspaces of equiangular rotations in 4D.