Let $A\in \mathbb{R}^{n\times n}$ be invertible such that $A$ and $A^{-1}$ have integer entries. Prove that $\det(A)\in\{-1,1\}$.
129647
$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$
9574
How to show that $\det(AB) =\det(A) \det(B)$?
39256
Finding $\det(I+A^{100})$ where $A\in M_3(R)$ and eigenvalues of $A$ are $\{-1,0,1\}$
52791
Invertibility of an operator $T$ if $\|I-T\|<1$
68724
$||T-I|| < 1$ implies that $T$ is invertible.
23224
$\mbox{rank}(A)=1$ implies $\det(A+I)=\mbox{trace}(A)+1$
32487
Proving $\det \left( \begin{smallmatrix} A & -B \\ B & A \end{smallmatrix} \right) =|\det(A+iB)|^2$
32108
$ a\in \Bbb Z_n$ is invertible $\,\Rightarrow\gcd (a,n) =1$
22215
How to prove Binet theorem, namely $\det ( AB ) = \det (A ) \det ( B )$?
64742
Prove a residue matrix $A$ (with coefficients in $\mathbb Z_n)$ has an inverse if and only if $\gcd(\det A,n) = 1$
55232
Show that $g\circ f\equiv 0$ implies $\det Df\equiv 0$, where $g(x_1,...,x_n)=x_1^5+...+x_n^5$
7034
Prove that $A+I$ is invertible if $A$ is nilpotent
107128
Let $B$ be a nilpotent $n\times n$ matrix with complex entries let $A = B-I$ then find $\det(A)$
148946
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
45658
Determinant of a special $n\times n$ matrix
25488
Let $A,B$ be $m\times m$ matrices such that $AB$ is invertible. Show $A,B$ invertible.
127127
Prove condition number of a invertible matrix is atleast one