Linked Questions

0
votes
1answer
129 views

Rank of matrices, prove inequality [duplicate]

Today I'm having hard time with linear algebra problems; this is one: $\forall A,B\in M_n(\mathbb{K})$, $\mathrm{rank}(A)+\mathrm{rank}(B)\le \mathrm{rank}(AB)+n$ $M_n(\mathbb{K})$ is the space of ...
43
votes
8answers
17k views

How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?

Let $A$ and $B$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$ I proved $\operatorname{rank}(AB) \...
1
vote
3answers
10k views

rank(AB) = rank(A) if B is invertible [duplicate]

If $B$ is invertible, show that rank($AB$) = rank($A$). I've seen this question asked elsewhere but all had answers I didn't understand. I know how to solve the following problem If $A$ is ...
7
votes
4answers
5k views

prove that $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$

If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that $$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$ Also show when equality occurs.
3
votes
3answers
571 views

Matrices and rank inequality

Let $A \in K^{m\times n}$ and $B \in K^{n \times r}$ Prove that min$\{rk(A),rk(B)\}\geq rk(AB)\geq rk(A)+rk(B)-n$ My attempt at a solution: $(1)$ $AB=(AB_1|...|AB_j|...|AB_r)$ ($B_j$ is the j-th ...
4
votes
1answer
3k views

How to show that $Rank(AB)\geq Rank(A)+Rank(B)-n$

Let $A\in M_{m \times n}$ and $B\in M_{n \times k}$. Prove that $$Rank(AB)\geq Rank(A)+Rank(B)-n.$$ I have tried to use $Im(AB) \subseteq Im(B)$ but that lead me to nowhere, how should I approach ...
0
votes
0answers
1k views

Proof of Sylvester rank inequality

I have to show that If $A$ and $B$ be two matrices of the same order $n$, then $$ \text{rank}\, A + \text{rank}\,B \le\text{rank}\,AB + n $$ In a previous exercise I have already shown that $$...
2
votes
2answers
173 views

For nonzeros $A,B,C\in M_n(\mathbb{R})$, $ABC=0$. Show $\operatorname{rank}(A)+\operatorname{rank}(B)+\operatorname{rank}(C)\le 2n$

Let $A,B,C\in M_n(\mathbb{R})$ be nonzero matrices such that $ABC=0$. How can we prove that $\operatorname{rank}(A)+\operatorname{rank}(B)+\operatorname{rank}(C)\le 2 n$ ? I can prove this for two ...
-1
votes
2answers
228 views

Let A be $ m \times n$ and $B$ be $n \times p$, and suppose $AB = 0$. Explain why $\mathrm{rank}(A) + \mathrm{rank}(B) \leq n.$ [closed]

Let $A$ be $m \times n$ and $B$ be $n \times p$, and suppose $AB = 0$. Explain why $\text{rank}(A) + \text{rank}(B) ≤ n$?
0
votes
2answers
197 views

Linear transformation over two vector spaces

We know linear transformation $T$ over two vector spaces $V,W$ and the rank $r$ (dimension of image $T$) of $T$ . We also know matrix representation $M$ of $T$. Is $r$ the rank of $M$ ?
0
votes
1answer
114 views

show that $\operatorname{rank}(g\circ f) \leq \operatorname{rank}(f)+\operatorname{rank}(g)-n$

Let $E$ a vector space and $\dim(E)=n$ and let $f,g \in L(E)$ show that $\operatorname{rank}(f\circ g) \leq \operatorname{rank}(f)+\operatorname{rank}(g)-n$ I can see that $\operatorname{Ker}(g) \...
-1
votes
3answers
164 views

Rank of matrices and their product

Let $\operatorname{rank}(A_{3 \times 3})=\operatorname{rank}(B_{3 \times 3})=2$. I need to figure out whether $AB=0$ is possible. On the one hand, I know that $\operatorname{rank}(AB) \leq \min(\...
2
votes
0answers
54 views

For $n\times n$ matrix $A$, there exists $B$ such that $AB=BA=0$ and $rk(A) + rk(B) = n$

For every $n\times n$ matrix $A$, there exists $B$ such that $AB=BA=0$ and $rk(A) + rk(B) = n$ The question is to prove or disprove this. I think it is true. Take a $n\times n$ matrix $\tilde A$ ...
1
vote
2answers
52 views

Question about elementary row operations with block matrices

Given two $n \times n$ matrices $A$ and $B$, form a new block matrix $$P := \begin{bmatrix}I_n&B\\-A&0\end{bmatrix}$$ Then by using only elementary row operations, show that $P$ can be ...