# Chapter 4: Area of a Parallelogram, Determinants, Volume and Hypervolume,
the Vector Product

##

## Introduction

We consider area of a parallelogram and volume of a parallelepiped and the
notion of determinant in two and three dimensions, whose magnitudes are these
for figures with their column vectors as edges. We then consider the application
of matrices to describing linear transformations on vectors, and methods for
evaluating determinants.

We further discuss the notion of the inverse of a matrix and how it can be
computed, and introduce the notions of eigenvalue and charactreristic equation,
and the
vector or cross product.

## Topics

4.1 Area, Volume and the Determinant in Two and Three
Dimensions

4.2 Matrices and Transformations on Vectors; the Meaning
of 0 Determinant

4.3 Evaluating the Determinant by Gaussian Elimination
and by Row or Column Expansion

4.4 The Determinant and the Inverse of a Matrix

4.5 The Vector Product

4.6 Eigenvalues and the Characteristic Equation of
a Matrix