Let $A$ be an $m\times n$ matrix with $\text{rank}(A) = 1$. Prove that for some column $u$ in $\mathbb{R}^m$ and row $v$ in $\mathbb{R}^n$, $A = uv$.
34043
A rank-one matrix is the product of two vectors
106584
If $A^{2n}=I$, then $A^n=\pm I$ where $A\in M_n(\mathbb{R})$
41436
rank of complex conjugate transpose matrix property proof
37162
Prove that matrices have equal rank.
150387
Questions about matrix rank, trace, and invertibility
87494
$\operatorname{rank}(T^3) + \operatorname{rank}(T) \geq 2\operatorname{rank}(T^2)$ for finite-dimensional $V$
8554
Proof: $\mathrm{adj}(\mathrm{adj}(A)) = (\mathrm{det}(A))^{n-2} \cdot A$ for $A \in \mathbb{R}^{n\times n}$
64644
An $r\times r$ submatrix of independent rows and independent columns is invertible (Michael Artin's book).
62065
Prove that $v_1, \dots v_n$ is a basis of V.
27474
A square matrix $n \times n$ is an invertible matrix iff the rank of the matrix is $n$.
84871
Determining the dimension of span$\{AB-BA : A,B \in M_{n \times n}(\mathbb R) \}$
34702
Show $|A + uv^T| = (1 + v^T A^{-1} u) |A|$
122853
Show that $\operatorname{rank}(A) = \operatorname{rank}(B)$
35897
Proof of: $AB=0 \Rightarrow Rank(A)+Rank(B) \leq n$
87441
Proof involving matrix equation
21758
Rank of the $n \times n$ matrix with ones on the main diagonal and $a$ off the main diagonal
123483
Prove that $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A + B)$