How do I prove that if $A$ is a $n\times n$ matrix with integer entries, then $A^{−1}$ has integer entries if and only if $\det(A) = ±1.$
129647
$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$
7205
Find the determinant of $A + I$, where $A$ is a real matrix such that $AA^{\top}=I$ and $\det A<0$.
35015
Why must the determinant of a matrix with with integer entries be an integer?
8554
Proof: $\mathrm{adj}(\mathrm{adj}(A)) = (\mathrm{det}(A))^{n-2} \cdot A$ for $A \in \mathbb{R}^{n\times n}$
45532
Let $A$ be a $n\times n$ matrix with entries $a_{ij}=i+j $ . Calculate rank of $A$
33477
Proving that if $A$ is a $8\times 8$ matrix over $\mathbb{R}$ and $A^3=A$, then $A$ is diagonalizable.
84522
A better proof for $\det(P) = \pm1$ if $P$ is an orthogonal matrix
78224
Suppose $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging two rows of $A$. Prove that $\det(B) = -\det(A)$
110271
Given a matrix with non-negative real entries, can you algebraically prove that it has a non-negative eigenvalue?
1865
Sub-determinants of an orthogonal matrix
150715
Prove that the determinant of skew-symmetric matrices of odd order is zero
69980
Proving that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$
80134
Maximum and minimum value of Determinant of $3 \times 3$ Matrix with entries $\pm 1$
134402
$\mathrm{rank}(A)+\mathrm{rank}(I-A)=n$ for $A$ idempotent matrix
106398
If $A$ is invertible show that $\det(A) \neq 0$
82925
Let the matrix $A=[a_{ij}]_{n×n}$ be defined by $a_{ij}=\gcd(i,j )$. How prove that $A$ is invertible, and compute $\det(A)$?
38436
If an orthogonal matrix has determinant -1 then it has -1 as an eigenvalue