inition of determinant of T, prove that det(T-¹) = (det(T))−¹ (Hir erminant of the identity man?)

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.9: Properties Of Determinants
Problem 34E
icon
Related questions
Question
100%
3) Suppose V is an n-dimensional F vector space and let T: V → V be a linear map.
(a)
Suppose that T is an isomorphism, and let T-1 denote its inverse. Using
our definition of determinant of T, prove that det (T-¹) = (det(T))-¹ (Hint: what is
the determinant of the identity map?)
(b)
Again using our definition of determinant, Show that T is an isomorphism
det(T) + 0. (Hint: for one direction use part a.
Transcribed Image Text:3) Suppose V is an n-dimensional F vector space and let T: V → V be a linear map. (a) Suppose that T is an isomorphism, and let T-1 denote its inverse. Using our definition of determinant of T, prove that det (T-¹) = (det(T))-¹ (Hint: what is the determinant of the identity map?) (b) Again using our definition of determinant, Show that T is an isomorphism det(T) + 0. (Hint: for one direction use part a.
Expert Solution
steps

Step by step

Solved in 4 steps with 4 images

Blurred answer
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning