If $ (A + B)^{-1} = A ^{-1} + B^{-1} $ then $\det A = \det B $.
73697
Let $A, B$ be $n\times n$ with $n\ge 2$ nonsingular matrices with real entries such that $A^{-1} + B^{-1} =(A+B)^{-1}$
131212
Prove that if $A$ is regular then $\operatorname{adj}(\operatorname{adj}(A)) = (\det A)^{n-2} A$
150309
Why do we assume $a+b=b+a$ in a Ring with 1. Also, is it true with Rings without 1?
4486
If $ar + bs =1$, then $\gcd(a,s) = \gcd(r,b) = \gcd(r,s) = 1$
126280
In a P.I.D., if $a^m = b^m$ and $a^n = b^n$ for $m, n \in \mathbb{N}$ with $\gcd(m,n) = 1$, then $a=b$
121767
Why $\det(A^{-1}) = 1/\det(A)$?
19457
Show that: $\inf(A+B) = \inf(A)+ \inf(B)$
101040
Prove the equality: $\det\left[\begin{smallmatrix} -2a &a+b &a+c \\ b+a&-2b &b+c \\ c+a&c+b &-2c \end{smallmatrix}\right] = 4(a+b)(b+c)(c+a)$
27281
How to prove $\,x^a-1 \mid x^b-1 \iff a\mid b$
87425
Prove that if $A$ is a square matrix with integer entries and $\det(A) = \pm 1$, then $A^{-1}$ contains all integer entries.
137626
If $a,b,c$ are positive integers,and $\cfrac{a}{b}+\cfrac{b}{c}+\cfrac{c}{a}$ and $\cfrac{b}{a}+\cfrac{c}{b}+\cfrac{a}{c}$ are integers then $a=b=c$
69721
$(a,b) \mathbin\# (c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$. Multiplicative inverse?
6128
If $a \cdot a = 1$ then $a = 1$ or $a = -1$
125030
Prove that $\det(A)=p_1p_2-ba={bf(a)-af(b)\over b-a}$
158862
Show that $a(-1) = (-1)a = -a $.
65923
How to approach proving $f^{-1}(B\setminus C)=A\setminus f^{-1}(C)$?
78224
Suppose $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging two rows of $A$. Prove that $\det(B) = -\det(A)$