query: If $A$ is invertible and $A$,$A^{-1}$ contains only integers, why is $\det(A^{4}) = 1$?
positive_passages.docid: 28861
positive_passages.text: Given an invertible matrix $A$ such that all elements in $A$ and in $𝐴^{−1}$ are integers, find $|𝐴^4|$
negative_passages.docid: 7034
negative_passages.text: Prove that $A+I$ is invertible if $A$ is nilpotent
negative_passages.docid: 158862
negative_passages.text: Show that $a(-1) = (-1)a = -a $.
negative_passages.docid: 63119
negative_passages.text: Let $M=\big( \begin{smallmatrix} A & B \\ C & D \end{smallmatrix} \big)$. Prove $\det(M)=\det(A)\cdot \det(D-C·A^{-1}·B)$.
negative_passages.docid: 96398
negative_passages.text: An integer $a \pmod m$ has inverse if and only if $\gcd(a,m)=1$?
negative_passages.docid: 13575
negative_passages.text: Need help for proving that: $f(f^{-1}(A)) ⊆ A$
negative_passages.docid: 101313
negative_passages.text: Minimal polynomial, determinants and invertibility
negative_passages.docid: 148946
negative_passages.text: If $\mathbf{A}$ is a $2\times 2$ matrix that satisfies $\mathbf{A}^2 - 4\mathbf{A} - 7\mathbf{I} = \mathbf{0}$, then $\mathbf{A}$ is invertible
negative_passages.docid: 115782
negative_passages.text: Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz
negative_passages.docid: 125030
negative_passages.text: Prove that $\det(A)=p_1p_2-ba={bf(a)-af(b)\over b-a}$
negative_passages.docid: 35015
negative_passages.text: Why must the determinant of a matrix with with integer entries be an integer?
negative_passages.docid: 98855
negative_passages.text: Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$
negative_passages.docid: 101452
negative_passages.text: How is the determinant related to the inverse of matrix?
negative_passages.docid: 98323
negative_passages.text: If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue?
negative_passages.docid: 81818
negative_passages.text: Prove or disprove : $\det(A^k + B^k) \geq 0$
negative_passages.docid: 33891
negative_passages.text: If $\mathbf A$ and $\mathbf B$ are Hermitian, when does $\det(\mathbf A-\lambda\mathbf B)=0$ have only real roots?
negative_passages.docid: 18108
negative_passages.text: Show if $(a,p)=1$ there is a unique inverse of $a$ modulo $p$