Let $A,B\in M_n(\mathbb{R})$, Prove $\det\begin{pmatrix} A & -B\\ B & A \end{pmatrix}=|\det(A+iB)|^2$
32487
Proving $\det \left( \begin{smallmatrix} A & -B \\ B & A \end{smallmatrix} \right) =|\det(A+iB)|^2$
145980
For every matrix $A\in M_{2}( \mathbb{C}) $ there's  $X\in M_{2}( \mathbb{C})$ such that $X^2=A$?
7384
How do I prove that $m(TE)=\left|\det(T)\right|m(E)$?
46780
Proving $((a\bmod n) (b\bmod n))\bmod n = ab\bmod n $
129647
$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$
89175
Proof of $(a+b)^{n+1}$
11895
If $AB=0$ prove that $\mathrm{rank}(A)+\mathrm{rank}(B)\leq n$
125348
Does $\det(A + B) = \det(A) + \det(B)$ hold?
118304
If $A^2 = I$ (Identity Matrix) then $A = \pm I$
72470
Prove that $2|ab| \leq a^2 + b^2$ and $|a|+|b| \leq \sqrt {2}(a^2+b^2)^{1/2}$
69980
Proving that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$
28565
Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$
146847
Given a square matrix A of order n, prove $\operatorname{rank}(A^n) = \operatorname{rank}(A^{n+1})$
23847
Determinant of a matrix whose $(i,j)$th element is $|i-j|$?
38378
$M= \{ A \in Mat_{2 \times 2}{\mathbb{R}}| \det(A)=1 \}$ is homeomorphic to $S^{1} \times \mathbb{R}^{2}$
128990
Prove that, at least one of the matrices $A+B$ and $A-B$ has to be singular
13026
Prove that if AB=I_m, then rk(A)=rk(B).