Guided Proof Prove that the determinant of aninvertible

Chapter 3, Problem 3.3.65

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Guided Proof Prove that the determinant of aninvertible matrix A is equal to 1 when all of the entriesof A and A1 are integers. Getting Started: Denote det(A) as x and det(A1) as y.Note that x and y are real numbers. To prove that det(A)is equal to 1, you must show that both x and y areintegers such that their product xy is equal to 1.(i) Use the property for the determinant of a matrixproduct to show that xy = 1.(ii) Use the definition of a determinant and the fact thatthe entries of A and A1 are integers to show thatboth x = det(A) and y = det(A1) are integers.(iii) Conclude that x = det(A) must be either 1 or 1because these are the only integer solutions to theequation xy = 1.