query: A problem on rank of $A$ and $\operatorname{adj}(A)$
positive_passages.docid: 4205
positive_passages.text: Is adjoint of singular matrix singular? What would be its rank?
negative_passages.docid: 81868
negative_passages.text: How to prove $\operatorname{Tr}(AB) = \operatorname{Tr}(BA)$?
negative_passages.docid: 55820
negative_passages.text: If $A$ is an $n \times n$ matrix and $ A^2 = 0$, then $\text{rank}(A)\le n/2$.
negative_passages.docid: 101079
negative_passages.text: Inner automorphisms form a normal subgroup of $\operatorname{Aut}(G)$
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: 5058
negative_passages.text: Eigenvalues of the rank one matrix $uv^T$
negative_passages.docid: 34043
negative_passages.text: A rank-one matrix is the product of two vectors
negative_passages.docid: 78575
negative_passages.text: If $A^2 = I$, then $\operatorname{rank} (A+I) + \operatorname{rank} (A-I) =n$
negative_passages.docid: 17430
negative_passages.text: If $\operatorname{rank}\left( \begin{bmatrix} A &B \\ C &D \end{bmatrix}\right)=n$ prove that $\det(AD)=\det(BC)$
negative_passages.docid: 59396
negative_passages.text: Show that the rank of $A+B$ is no more than the sum of the ranks of $A$ and $B$
negative_passages.docid: 110018
negative_passages.text: Conjugacy classes in $A_n$.
negative_passages.docid: 153823
negative_passages.text: $|\text{rank}A-\text{rank}B|\leq \text{rank}(A+B)$
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: 97487
negative_passages.text: How to prove whether a matrix has rank $1$
negative_passages.docid: 147659
negative_passages.text: Counterexample to "if $f:A\to A$ induces the identity $\hat f:\operatorname{Spec}(A)\to\operatorname{Spec}(A)$, then $f=\operatorname{id}_A$"
negative_passages.docid: 6905
negative_passages.text: Rank of product of a matrix and its transpose
negative_passages.docid: 89177
negative_passages.text: $\operatorname{adj}(AB) = \operatorname{adj} B \operatorname{adj} A$