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

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

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.
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