Show that $\mathrm{rank}(I_n-A^TA)-\mathrm{rank}(I_m-AA^T) \geq n-m.$
123366
Determinant of block matrices with non-square blocks
117688
Prove that $\def\Aut{\operatorname{Aut}}\Aut(\mathbf{Z_{n}})\simeq \mathbf{Z_{n}^{*}}$
7843
Show that order of $a^k$
36747
$\text{Tor}_0^R(M,N)\simeq M\otimes_R N$
145752
Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$
23224
$\mbox{rank}(A)=1$ implies $\det(A+I)=\mbox{trace}(A)+1$
77192
How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
87494
$\operatorname{rank}(T^3) + \operatorname{rank}(T) \geq 2\operatorname{rank}(T^2)$ for finite-dimensional $V$
14646
Prove $\det(kA)=k^n\det A$
118471
Prove that $1-\sum_{i=1}^n a_i < \prod_{i=1}^n(1-a_i) $
96152
Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$
33903
Prove that $(1+a_1) \cdot (1+a_2) \cdot \dots \cdot (1+a_n) \geq 2^n$
89325
$a_{m+n}\leq a_ma_n$ implies the convergence of $\sqrt[n]a_n$
68705
Prove that $A\ge0, B\ge0$ and $A\ge B$ implies $B^{-1}\ge A^{-1}$
1478
An inequality for powers of reals: $a^{1/n}-b^{1/n}<(a-b)^{1/n}$
11895
If $AB=0$ prove that $\mathrm{rank}(A)+\mathrm{rank}(B)\leq n$
97487
How to prove whether a matrix has rank $1$