query: Let $A$ and $B$ be $n\times n$ matrices. Let $\operatorname{rank}(A)=s$ and $\operatorname{rank}(B)=t$. Then rank of $A+B\ge\cdots$
positive_passages.docid: 153823
positive_passages.text: $|\text{rank}A-\text{rank}B|\leq \text{rank}(A+B)$
negative_passages.docid: 126672
negative_passages.text: Product of principal ideals: $(a)\cdot (b) = (a b)$
negative_passages.docid: 45469
negative_passages.text: If $a_k = \operatorname{rank}(S^k)$, deduce that $a_k - a_{k+1} \geq a_{k+1} - a_{k+2}$
negative_passages.docid: 111031
negative_passages.text: Matrix rank and linear independence of functions
negative_passages.docid: 25648
negative_passages.text: Prove $(a_1+b_1)^{1/n}\cdots(a_n+b_n)^{1/n}\ge \left(a_1\cdots a_n\right)^{1/n}+\left(b_1\cdots b_n\right)^{1/n}$
negative_passages.docid: 27821
negative_passages.text: A problem on rank of $A$ and $\operatorname{adj}(A)$
negative_passages.docid: 39585
negative_passages.text: Connection between rank and matrix product
negative_passages.docid: 33560
negative_passages.text: Show that matrices $A$ and $B$ commute if and only if $\begin{vmatrix} b & a-c \\ e & d-f\end{vmatrix}=0.$
negative_passages.docid: 27248
negative_passages.text: If $A$ is an $m \times n$ matrix, prove that $Ax=0$ has infinite solutions $\iff$ $\text{rank}(A)<n$
negative_passages.docid: 30866
negative_passages.text: Relation between rank and number of distinct eigenvalues of a matrix
negative_passages.docid: 97487
negative_passages.text: How to prove whether a matrix has rank $1$
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: 119435
negative_passages.text: Rank of the difference of matrices
negative_passages.docid: 151443
negative_passages.text: Find two $2 \times 2$ matrices $A$ and $B$ with the same rank, determinant,...but they are not similar.
negative_passages.docid: 67058
negative_passages.text: Prove that $\text{rank}(X^TX)=\text{rank}(X)$
negative_passages.docid: 128760
negative_passages.text: If $A$ and $B$ two $n\times n$ matrix such that $AB=BA$, then $\dim(A\cap B)$?
negative_passages.docid: 87387
negative_passages.text: Linear equations given by $Ax = b$ have a solution if and only if $\operatorname{rank}(A|b) = \operatorname{rank}(A)$