query: Proof that there is a unique solution to $Ax=b$ implies rank = $n$ of a $m \times n$ matrix $A$
positive_passages.docid: 87387
positive_passages.text: Linear equations given by $Ax = b$ have a solution if and only if $\operatorname{rank}(A|b) = \operatorname{rank}(A)$
negative_passages.docid: 158845
negative_passages.text: Given $A$ and $B$ positive-definite matrices and $Q$ unitary matrix, prove that if $A = BQ$, then $A=B$.
negative_passages.docid: 138025
negative_passages.text: How to show the solution of $\dot{x} = Ax $ is an invariant subspace?
negative_passages.docid: 45532
negative_passages.text: Let $A$ be a $n\times n$ matrix with entries $a_{ij}=i+j $ . Calculate rank of $A$
negative_passages.docid: 13026
negative_passages.text: Prove that if AB=I_m, then rk(A)=rk(B).
negative_passages.docid: 32155
negative_passages.text: Prove that if for every $x \in \mathbb{R}^N$ $Ax=Bx$ then $A=B$
negative_passages.docid: 75211
negative_passages.text: Theorem about positive matrices
negative_passages.docid: 128990
negative_passages.text: Prove that, at least one of the matrices $A+B$ and $A-B$ has to be singular
negative_passages.docid: 112428
negative_passages.text: Existence of least squares solution to $Ax=b$
negative_passages.docid: 98298
negative_passages.text: Prove that bilinear form can be presented as product of linear forms if and only if it has rank one
negative_passages.docid: 75117
negative_passages.text: $\mathrm{rank}(AB-BA)=1$ implies $A$ and $B$ are simultaneously triangularisable
negative_passages.docid: 140446
negative_passages.text: Prove that $A \smallsetminus (A \smallsetminus B) = A \cap B$
negative_passages.docid: 50266
negative_passages.text: In the following figure prove that: $AK||BC$
negative_passages.docid: 65783
negative_passages.text: Proving: "The trace of an idempotent matrix equals the rank of the matrix"
negative_passages.docid: 27474
negative_passages.text: A square matrix $n \times n$ is an invertible matrix iff the rank of the matrix is $n$.
negative_passages.docid: 116823
negative_passages.text: VERIFYING: Proving an $n \times n$ Matrix Vector Space
negative_passages.docid: 77192
negative_passages.text: How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?