Prove that $\det A = 1$ with $A^T M A = M$ and $M = \begin{bmatrix} 0 & I \\ -I &0 \end{bmatrix}$.
34445
Why is the determinant of a symplectic matrix 1?
101040
Prove the equality: $\det\left[\begin{smallmatrix} -2a &a+b &a+c \\ b+a&-2b &b+c \\ c+a&c+b &-2c \end{smallmatrix}\right] = 4(a+b)(b+c)(c+a)$
69980
Proving that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$
38378
$M= \{ A \in Mat_{2 \times 2}{\mathbb{R}}| \det(A)=1 \}$ is homeomorphic to $S^{1} \times \mathbb{R}^{2}$
28565
Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$
144221
Matrix determinant lemma with adjugate matrix
116376
Prove the determinant is the product of its diagonal entries
23095
Prove that T is a linear transformation
125030
Prove that $\det(A)=p_1p_2-ba={bf(a)-af(b)\over b-a}$
109745
If $(a,k)=(b,m)=1$, prove that $(ab,km)=(a,m)(b,k)$.
20110
How to prove $(AB)^T=B^T A^T$
130498
Linear algebra proof regarding matrices
152932
Proof: $\det\pmatrix{\langle v_i , v_j \rangle}\neq0$ $\iff \{v_1,\dots,v_n\}~\text{l.i.}$
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 $\det(A)$?
29462
Suggestions on how to prove the following equality. $a^{m+n}=a^m a^n$
129647
$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$
7712
How to prove the both identity (matrix)