PLEASE SHOW EACH AND EVERY STEP. THANK YOU!!! 5 points] a) Suppose A is a a nons

Question

PLEASE SHOW EACH AND EVERY STEP. THANK YOU!!!

5 points] a) Suppose A is a a nonsingular nx n matrix with integer entries, and suppose A-1 also has integer entries. Show det(A)-L. Hint: det(A-1)-det(A) i, and the determinant of a matrix with integer entries is an integer. The reciprocal of an integer n is an integer if and only if n1 b) Conversely, if A has integer entries and det A 1, show that A-1 also has integer entries. Hint: Use the equation that relates A-1 and adj(A) to reduce this to showing that adj(A) has integer entries, and then use part of the hint for la).

Explanation / Answer

a) det(A) is non zero as A is non singular

But, A and A^{-1} both have integer entries hence, det(A) is an integer

det(A^{-1})=1/det(A) is also an integer as A^{-1} also has integer entries

Hence, det(A)=+-1

b)

A^{-1}=cof(A)/det(A)

cof(A) is the cofactor matrix constructed using A

Each cofactor is entry is a sum of product of elements of A hence each entry in cof(A) is an integer and det(A)=+-1

Hence, each entry in inverse of A is also an integer

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