Let A be an invertible matrix all of whose coefficients are integers. Which of the following statements is NOT true for all possible matrices A?

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To evaluate which statement is NOT true for all invertible integer matrices A, let's examine each option carefully.
Option a: 'All coefficients of the matrix det(A) A^{-1} are integers.' Since A has integer entries, det(A) A^{-1} equals the adjugate matrix adj(A), whose entries are the cofactors of A. Cofactors are polynomials with integer coefficients in the entries of A, so adj(A) has integer entries. Therefore det(A) A^{-1} has only integers for every integer matrix A, and this statement is true.
Option b: 'All coefficients of the matrix A^{-1} are integers.' This is not guaranteed. A counterexample suffices: take A = [[2, 0], [0, 3]]. Then A^{-1} = [[1/2, 0], [0, 1/3]], which clearly has non-integer entries. Since there exists at least one invertible integer matrix with a non-integer inverse, this statement is not true for all A.
Option c: 'AA^{-1} only has integer coefficients.' By definition, AA^{-1} = I, the identity matrix. The identity has 1s on the diagonal and 0s elsewhere, all integers. Hence this statement is true for all invertible integer matrices A.
Summarizing the reasoning: option b is the only statement that fails universally, because the inverse of a general integer matrix need not have integer entries, while det(A) A^{-1} recovers the integer adjugate and AA^{-1} yields the integer identity. Therefore, the statement NOT true for all A is option b.

MTH1030 -1035 - S1 2025 MTH1030/5 Week 3 lesson quiz: Matrices and determinants

Let A be an invertible matrix all of whose coefficients are integers. Which of the following statements is NOT true for all possible matrices A?

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