MATH-332: Linear Algebra 1-1
MATH-332: Linear Algebra Chapter: 3
Determinants
Section 3.3: Cramer’s Rule, Volume, Linear Transformations
pgs. 201 - 209 July 3, 2009
Lecture: Cramer’s Rule, Volume, Linear Transformations
Topics:
Cramer’s Rule
An Inverse Formula
Determinants as Volumes
Linear Transformations in R2,R3
Problems Prac: 1
Prob: 5, 7, 13, 17, 19, 25
Section Goals
•Understand the proof of Cramer’s Rule and how this theorem can be used to construct an
element level description of inverse of a nonsingular matrix.
•Interpret the determinant geometrically as the volume of the parallelepiped induced by the
vectors, which make up the columns of a matrix.
Section Objectives
•Prove Cramer’s Rule, theorem 3.7 page 201, through the use of properties of determinants thus
giving an element level description of the solution to a square linear system with nonsingular
coefficient data.
•Using Cramer’s rule and the definition of matrix products to prove theorem 3.8, which gives
an element level description of an inverse matrix.
•State the implications of theorem 3.9, which says that the determinant is related to a unsigned
spanned area use this and theorem 3.10 on page 207 to discuss how the determinant can be
used to calculate the way linear transformations change areas.