Let A be a nonsingular square matrix with integer entries. Prove that inverse of A contains only integer entries if and only if det(A) = ±1.
87425
Prove that if $A$ is a square matrix with integer entries and $\det(A) = \pm 1$, then $A^{-1}$ contains all integer entries.
37249
Suppose $A$ is a 4x4 matrix such that $\det(A)=\frac{1}{64}$
124942
Prove that an upper triangular matrix is invertible if and only if every diagonal entry is non-zero.
63119
Let $M=\big( \begin{smallmatrix} A & B \\ C & D \end{smallmatrix} \big)$. Prove $\det(M)=\det(A)\cdot \det(D-C·A^{-1}·B)$.
7205
Find the determinant of $A + I$, where $A$ is a real matrix such that $AA^{\top}=I$ and $\det A<0$.
67784
If $A + A^t = 2I$ then $\det(A) \geq 1$
121767
Why $\det(A^{-1}) = 1/\det(A)$?
124149
Inverse of matrix with nonnegative entries
115296
Under what conditions does a matrix $A$ have a square root?
97487
How to prove whether a matrix has rank $1$
133449
A proof that an orthogonal matrix with a determinant 1 is a rotation matrix
98631
Postive-semidefiniteness of matrix with entries $1/(a_i+a_j)$
131480
Proof that composition of invertible linear transformations is invertible (without determinants)
75002
show that $\det(A)=0$ in this case
125889
Determinant of rank-one perturbations of (invertible) matrices
151526
Determinant of a matrix with different cases for its entries
517
Determinant of a specially structured matrix ($a$'s on the diagonal, all other entries equal to $b$)