These are some questions based on Rank Of a matrix!!!

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February 16th 2013, 03:44 AM
tejasnatu

These are some questions based on Rank Of a matrix!!!

Let A and B be n*n matrices each having rank n. Then rank of C = A + B is
also n.. Is this True or False.. Give Reasons???

Let A = (aij) be a n*n matrix where aij = (i + j). Find the rank of A..
February 16th 2013, 06:53 AM
zhandele

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Re: These are some questions based on Rank Of a matrix!!!

Instinctively I'd say no. I found a counterexample, though it might seem a little contrived. See the attached pdf. The two 4x4 matrices both have rank 4, but their sum has rank 3.

It's true that if you generate a random nxn matrix, its rank will be n. You could generate random matrices in pairs, add them, take the rank of the sum, repeat this all day, you'd almost certainly find that every sum had a rank of n.

You seem to ask two questions, but it seems to me that the second question is just the first question in a different form.
February 16th 2013, 07:30 AM
ILikeSerena

Re: These are some questions based on Rank Of a matrix!!!

Hi tejasnatu! :)

$\begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix}1 & 1 \\ 1 & 1 \end{pmatrix}$

What are their ranks?
February 16th 2013, 07:34 AM
tejasnatu

Re: These are some questions based on Rank Of a matrix!!!

hey thats rite... Rank of both the matrices on the left is 2, nd on the right is 1.. so the statement is false..
February 16th 2013, 07:35 AM
ILikeSerena

Re: These are some questions based on Rank Of a matrix!!!

Yep!
February 16th 2013, 11:55 AM
Deveno

Re: These are some questions based on Rank Of a matrix!!!

suppose A has rank n. clearly -A ALSO has rank n.

however, A + -A has rank 0.
February 16th 2013, 04:44 PM
tejasnatu

Re: These are some questions based on Rank Of a matrix!!!

And I believe the rank of A in the second question is 1..
February 16th 2013, 04:46 PM
ILikeSerena

Re: These are some questions based on Rank Of a matrix!!!

Quote:

Originally Posted by tejasnatu

Let A = (aij) be a n*n matrix where aij = (i + j). Find the rank of A..

Quote:

Originally Posted by tejasnatu

And I believe the rank of A in the second question is 1..

Let's see, suppose we take n=2:

$A = \begin{pmatrix}2 & 3 \\ 3 & 4 \end{pmatrix}$

What is its rank?
February 16th 2013, 08:01 PM
tejasnatu

Re: These are some questions based on Rank Of a matrix!!!

Well I am so sorry, I made a blunder wen i wrote the rank in the second question is 1. Actually I checked and found that rank for 2*2, 3*3, 4*4, 5*5 matrices remains the same equal to 2. M trying to come up with a generalised proof. Can you suggest something on that..
February 16th 2013, 08:30 PM
tejasnatu

Re: These are some questions based on Rank Of a matrix!!!

Well its certain that the rank is 2. I have proved it for any generalised n*n matrix. Thank you Ilikeserena for your interest in these problems. Looking forward for some more mathematical interaction...!!!