query: $\det(X+A)=\det X+ \text{tr}(X^*A)$
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: 51800
negative_passages.text: 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 $\det(AA^T)=nk^2.$
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: 69980
negative_passages.text: Proving that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$
negative_passages.docid: 152932
negative_passages.text: Proof: $\det\pmatrix{\langle v_i , v_j \rangle}\neq0$ $\iff \{v_1,\dots,v_n\}~\text{l.i.}$
negative_passages.docid: 60598
negative_passages.text: Matrix $A^T A$ as sum of outer products
negative_passages.docid: 121767
negative_passages.text: Why $\det(A^{-1}) = 1/\det(A)$?
negative_passages.docid: 131408
negative_passages.text: Find the expansion for $\det(I+\epsilon A)$ where $\epsilon$ is small without using eigenvalue.
negative_passages.docid: 133847
negative_passages.text: Derivative of determinant of a matrix
negative_passages.docid: 28565
negative_passages.text: Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$
negative_passages.docid: 75002
negative_passages.text: show that $\det(A)=0$ in this case
negative_passages.docid: 141707
negative_passages.text: Compute the derivative of the log of the determinant of A with respect to A
negative_passages.docid: 80015
negative_passages.text: $Tr(A^2)=Tr(A^3)=Tr(A^4)$ then find $Tr(A)$
negative_passages.docid: 52973
negative_passages.text: The identity $(u\times v)\cdot(x\times y)=\begin{vmatrix}u\cdot x&v\cdot x\\ u\cdot y&v\cdot y\\\end{vmatrix}$
negative_passages.docid: 125030
negative_passages.text: Prove that $\det(A)=p_1p_2-ba={bf(a)-af(b)\over b-a}$
negative_passages.docid: 81818
negative_passages.text: Prove or disprove : $\det(A^k + B^k) \geq 0$
negative_passages.docid: 103061
negative_passages.text: Why is it true that $\mathrm{adj}(A)A = \det(A) \cdot I$?