If $A$ is an $n×n$ orthogonal matrix with $\text{det}(A)=1$ and $n$ odd, then show that $\text{det}(I-A)=0$.
101926
If $Q$ is a proper orthogonal transformation matrix, deduce that $\det(1-Q)=0$.
27248
If $A$ is an $m \times n$ matrix, prove that $Ax=0$ has infinite solutions $\iff$ $\text{rank}(A)<n$
75855
Show that $\det{(A + sxy^*)}$ is linear in $s$
106398
If $A$ is invertible show that $\det(A) \neq 0$
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 $\det(A)$?
51800
We have matrix $A\in M_{n-1\times n}(\mathbb Z)$ so that the sum of entries in each row is zero. Prove that $\det(AA^T)=nk^2.$
69474
Orthogonal projections with $\sum P_i =I$, proving that $i\ne j \Rightarrow  P_{j}P_{i}=0$
142980
If $\,A^k=0$ and $AB=BA$, then $\,\det(A+B)=\det B$
60723
Prove that if $A$ is normal, then eigenvectors corresponding to distinct eigenvalues are necessarily orthogonal (alternative proof)
8554
Proof: $\mathrm{adj}(\mathrm{adj}(A)) = (\mathrm{det}(A))^{n-2} \cdot A$ for $A \in \mathbb{R}^{n\times n}$
41514
Let $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?
37537
Show that $det(A)=0$ for a Matrix $A$ Such That for $X=[2,3,4]^{T}$, $AX=0$
84522
A better proof for $\det(P) = \pm1$ if $P$ is an orthogonal matrix
36771
How to prove that A is normal if and only if there's a unitary matrix V,s.t. A*=AV?
99600
Show that a 2x2 matrix A is symmetric positive definite if and only if A is symmetric, trace(A) > 0 and det(A) > 0
57771
Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$.
124095
$A$ is a real orthogonal matrix, prove that $ (I+A)^{-1}(I-A)$ is skew-symmetric