What's the trick for proving one eigenvalue of orthogonal matrix is $-1$ if the determinant is $-1$?
38436
If an orthogonal matrix has determinant -1 then it has -1 as an eigenvalue
115467
Eigenvalues of Householder matrix
75002
show that $\det(A)=0$ in this case
86940
Proving an inequality in Eigenvalues of a matrix
60723
Prove that if $A$ is normal, then eigenvectors corresponding to distinct eigenvalues are necessarily orthogonal (alternative proof)
88473
Proof that the trace of a matrix is the sum of its eigenvalues
50010
Is this how eigenvalues of some matrix $A$ are related to the inverse of $A$?
81275
Eigenvalue Problem -- prove eigenvalue for $A^2 + I$
101926
If $Q$ is a proper orthogonal transformation matrix, deduce that $\det(1-Q)=0$.
129977
Prove that $\lambda_1^2$, $\lambda_1\lambda_2$ and $\lambda_2^2$ are eigenvalues of matrix $A$
104293
How to prove "eigenvalues of polynomial of matrix $A$ = polynomial of eigenvalues of matrix $A$ "
127127
Prove condition number of a invertible matrix is atleast one
54740
Proofs of Determinants of Block matrices
86125
If $A$ is idempotent, then the eigenvalues of $A$ are $0$ or $1$
2271
Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix
6960
Necessary and sufficient condition for the matrix $A = I - 2 x x^t$ to be orthogonal
19649
A question about the proof of "a symmetric matrix has real eigenvalues".