Guided Proof Prove that the determinant of aninvertible matrix A is equal to 1 when all

Chapter 3, Problem 65

(choose chapter or problem)

Prove that the determinant of an invertible matrix A is equal to \(\pm 1\) when all of the entries of A and \(A^{-1}\) are integers.

Getting Started: Denote det(A) as x and \(\operatorname{det}\left(A^{-1}\right)\) as y. Note that x and y are real numbers. To prove that det(A) is equal to \(\pm 1\), you must show that both x and y are integers such that their product xy is equal to 1.

(i) Use the property for the determinant of a matrix product to show that xy = 1.

(ii) Use the definition of a determinant and the fact that the entries of A and \(A^{-1}\) are integers to show that both x = det(A) and \(y=\operatorname{det}\left(A^{-1}\right)\) are integers.

(iii) Conclude that x = det(A) must be either 1 or −1 because these are the only integer solutions to the equation xy = 1.

Text Transcription:

pm 1

A^-1

det(A^-1)

y=det(A^-1)