3 Let A be an n n square matrix with integer entries and su

(3) Let A be an n × n square matrix with integer entries and such that det A does not equal to 0.

(a) Prove that then A^(1) has rational entries.

(b) Give an example when A^(1) is not an integer matrix .

(c) Prove that if det(A) = 1, then A^(1) is an integer matrix.

Solution

( a ) Suppose that A has integer entries such that det( A ) 0. Then Adj(A) being the matrix of cofactors of A will also have integer entries. We know that det(A^-1 A) = det( I_n ) = 1.Also, det(A^-1A)= det(A^-1)det(A). Therefore, det(A^-1)det(A) = 1 or, det (A^-1)= 1/det (A). Since det(A) is an integer when A has all integer entries, therefore det (A^-1) being the reciprocal of an integer will be a rational number. Therefore A^-1 = Adj (A)/ det (A) will have rational entries. ( as the product of an integer and a rational number is a rational number).

( b ) Let A 2 matrix with ( 3 , 2) as the 1^st row and ( 1, 3) as the 2^nd row. Then the 1^st row of A^-1 is ( 3/7, -2/7) and the 2^nd row is ( -1/7, 3/7). We may observe that det ( A) = 13 0

( c ) We know that A^-1 = Adj( A) /det (A) where Adj(A) is the matrix of co factors of A. If A has all integer entries, then Adj (A) will also have all integer entries since the cofactors of A will be all integers. Thus, if det (A) = 1, then A^-1 = Adj( A). Thus A^-1    has all integer entries.