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?

Question

BlackTom AI · Accepted Answer

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.

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?单项选择题

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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?单项选择题

类似问题

If 𝑀 = [ 1 − 5 − 3 0 1 1 0 0 1 ] , then which of the following is 𝑀 − 1 ?

如果 M = \left [\bestart{matrix}1 & 0 & -1\\0 & 1 & 7\\0 & 0 & 1\end{matrix}\right ]， 那么以下哪一个是\（M^{-1}\）？

Let \( A = \begin{bmatrix} 2 & -1 \\ -3 & -4 \end{bmatrix}\).Which of the following statements is correct?

Let \( A = \begin{bmatrix} 2 & -1 \\ -3 & -4 \end{bmatrix}\).Which of the following statements is correct?

The inverse of the matrix ( 1 0 1 4 2 0 𝑎 0 𝑎 2 ) is ( − 1 0 0.5 2 0.5 − 1 𝑎 2 0 − 0.5 ) . What is 𝑎 ?

If M=[1 0 0 0 1 0 6 3 1], then which of the following is M−1?

If \[M = \left [\begin{matrix}1 & 0 & 0\\-9 & 1 & 0\\6 & 2 & 1\end{matrix}\right ],\] then which of the following is \(M^{-1}\)?

更多留学生实用工具

希望你的学习变得更简单

如果 M = \left [\bestart{matrix}1 & 0 & -1\\0 & 1 & 7\\0 & 0 & 1\end{matrix}\right ]，那么以下哪一个是\（M^{-1}\）？