query: Does $\det(I+A) = 1 + \mbox{tr}(A)$ hold if $A$ is a rank-$1$ complex matrix?
positive_passages.docid: 2271
positive_passages.text: Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix
negative_passages.docid: 8554
negative_passages.text: Proof: $\mathrm{adj}(\mathrm{adj}(A)) = (\mathrm{det}(A))^{n-2} \cdot A$ for $A \in \mathbb{R}^{n\times n}$
negative_passages.docid: 65783
negative_passages.text: Proving: "The trace of an idempotent matrix equals the rank of the matrix"
negative_passages.docid: 101452
negative_passages.text: How is the determinant related to the inverse of matrix?
negative_passages.docid: 46825
negative_passages.text: The identity det(A) = exp(Tr(ln(A)) for A general
negative_passages.docid: 106305
negative_passages.text: Prove/disprove: if $\det(A+X) = \det(B + X)$ for all $X$, then $A=B$
negative_passages.docid: 137422
negative_passages.text: $\mathrm{card}(\mathbb{Z}^n/M\mathbb{Z}^n) = |\det(M)|$?
negative_passages.docid: 7205
negative_passages.text: Find the determinant of $A + I$, where $A$ is a real matrix such that $AA^{\top}=I$ and $\det A<0$.
negative_passages.docid: 122951
negative_passages.text: If the $\det(A)=0$ why does the matrix $A$ have an eigenvector?
negative_passages.docid: 63119
negative_passages.text: Let $M=\big( \begin{smallmatrix} A & B \\ C & D \end{smallmatrix} \big)$. Prove $\det(M)=\det(A)\cdot \det(D-C·A^{-1}·B)$.
negative_passages.docid: 19203
negative_passages.text: Is the rank of a matrix the same of its transpose? If yes, how can I prove it?
negative_passages.docid: 133449
negative_passages.text: A proof that an orthogonal matrix with a determinant 1 is a rotation matrix
negative_passages.docid: 85167
negative_passages.text: When A and B are of different order given the $\det(AB)$,then calculate $\det(BA)$
negative_passages.docid: 31116
negative_passages.text: Matrix given by $a_{ij} = 1/(i+j)$ is non-singular.
negative_passages.docid: 5058
negative_passages.text: Eigenvalues of the rank one matrix $uv^T$
negative_passages.docid: 93903
negative_passages.text: What is the determinant of cofactor matrix of a matrix?
negative_passages.docid: 67784
negative_passages.text: If $A + A^t = 2I$ then $\det(A) \geq 1$