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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