How to prove that if $A$ and $B$ are $n\times n$ matrices then $\operatorname{rank}(A) + \operatorname{rank}(B) \le \operatorname{rank}(AB) + n$
33451
Sylvester rank inequality: $\operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n$
33448
Proof if $AB+BA=0$ Then atleast one of the matrices are singular.
55820
If $A$ is an $n \times n$ matrix and $ A^2 = 0$, then $\text{rank}(A)\le n/2$.
63535
Prove that if $A – C = B – C$ and $A\cap C = B\cap C$ then $A = B$
27248
If $A$ is an $m \times n$ matrix, prove that $Ax=0$ has infinite solutions $\iff$ $\text{rank}(A)<n$
19203
Is the rank of a matrix the same of its transpose? If yes, how can I prove it?
123044
Commutative matrices are multiples of the identity
97487
How to prove whether a matrix has rank $1$
146847
Given a square matrix A of order n, prove $\operatorname{rank}(A^n) = \operatorname{rank}(A^{n+1})$
41026
How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?
113474
An $n \times n$ matrix $M$ of integers has the property that $M_{ab} = M_{xy}$ iff $a+b=x+y$. How many distinct elements are there in $M$?
10483
Prove Sylvester rank inequality: $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n$
59396
Show that the rank of $A+B$ is no more than the sum of the ranks of $A$ and $B$
17430
If $\operatorname{rank}\left( \begin{bmatrix} A &B \\ C &D \end{bmatrix}\right)=n$ prove that $\det(AD)=\det(BC)$
75966
Prove that if $A$ is a set with $m$ elements, $B$ is a set with $n$ elements, and $A \cap B = \emptyset$, then $A \cup B$ has $m+n$ elements.
94639
Proof that the rank of a skew-symmetric matrix is at least $2$
131212
Prove that if $A$ is regular then $\operatorname{adj}(\operatorname{adj}(A)) = (\det A)^{n-2} A$