The rank of the Sum of two matrices

1. May 31, 2016

Dank2

1. The problem statement, all variables and given/known data
Let A,B be square matrices of order n. n>=2
lets A and B be matrices of Rank 1. What are the options of the Rank of A+B ?

2. Relevant equations

3. The attempt at a solution
I know that there are 3 possibilities, 2, 1 , 0. Just having trouble with coming up with a formula. i tried:
Rank(A+B) = dim(SP{Ac+Bc}) , (where Ac is the columns of A.)
I know also this equation
Rank of (A+B) = n - dim(P(A+B)) (where P denotes the solution space of (A+B)x = 0.

2. May 31, 2016

Staff: Mentor

There is no formula. For all three cases there can easily be found an example. How did you show that it cannot be more than two?

3. May 31, 2016

Dank2

let Ac denote the span of the columns of Matrix A.
A+B is contained in Ac+Bc
therefore, dim(A+B) is smaller or equal to dim(Ac+Bc)

but dim(Ac+Bc) = dim(Ac) + dim(Bc) - dim(Ac intersection Bc)

4. May 31, 2016

Dank2

Fixed typing errors.

5. May 31, 2016

Staff: Mentor

Yes. But your A+B should both be (A+B)c and maybe a ≤ 1+1=2 at the end.
If you give examples for the three cases that is all you can do.

6. May 31, 2016

Dank2

i wanted to use Ac + Bc, because then, i can use the fact that Ac is a vector space, and usehe sum of the dimensions formula of vector spaces, which is dim(A+B)= dim(A)+dim(B) - dim(AintersectB)

7. May 31, 2016

Dank2

I was thinking if i could show it with nxn general matrices, but i can't think about a way,

8. May 31, 2016

Staff: Mentor

Two by two are enough. But you can attach zeros as many as you want or other linear dependencies. But why? A minimal example will be fine.

9. Jun 4, 2016

Dank2

I have been told a numerated example is not enough.I need to show it the possibilities in general. i've shown how the Rank of A + B is less or equal to 2. not sure if that's the correct way.

10. Jun 4, 2016

Staff: Mentor

Beside what I've said in post #5 about A+B (which should be (A+B)c) and the unusual notation for a linear span by c it is ok.
What does it mean for a (n,n)-matrix to have rank 0,1 or 2?

11. Jun 4, 2016

Dank2

not sure if i got the question right.
Rank = number of linearly independent vectors in columns or A or the the rows of A.

or that the columns of A span the 0 vector in Rn, a line in Rn or a plain in Rn respectively.

12. Jun 4, 2016

Staff: Mentor

Let's say rows. (Columns would work as well.)
So rank 0 means no linear independent row. That leaves only one possible vector.
Rank 1 thus means exactly one linear independent row. Let us take any fixed row vector as first. What does it mean for all others?
And at last two linear independent rows, but only two. All others must be in their span.
If you've found 3 matrices C with these properties, then you will have found C = A+B. All it needs then is to find summands A, B which add up to your C.