Let A be a square matrix all of whose entries are integers. Then which one of the following is true? (1) If `d e t A""=+-1,""t h e n""A^(1)` exists but all its entries are not necessarily integers (2) If `d e t A!=""+-1,""t h e n""A^(1)` exists and all its entries are non-integers (3) If `d e t A""=+-1,""t h e n""A^(1)` exists and all its entries are integers (4) If `d e t A""=+-1,""t h e n""A^(1)` need not exist

Options

A. If `det A ne 1,` then `A^(-1)` exists and all its entries are non-integers
B. If `det A = pm 1. ` then `A^(1)` then `A^(-1)` exist and all its entries are integers
C. If `det A = pm 1, ` then `A^(-1)` need not exist
D. If `det A = pm 1, ` then ` A^(-1)` exists but all its entries are not
necessarily integers

Answer

D

Explanation

Let `A= [[2,1],[0,1//2]]`
`detA= abs[[2,1],[0,1//2]]=1`
and `A^(-1) = [[1//2,-1],[0,2]]`
and let `A= [[3,0],[-3,-1//3]],`
`detA=abs((3,0),(-3,-1//3))=-1`
and `A^(-1)= [[1//3,0],[-3,-3]]`