does the elementary transformation to a matrix change the rank?

Asked Oct 08 '15 at 02:18

Active Oct 08 '15 at 02:18

Viewed 7,247 times

Does the Rank decrease if the elements of a row are reduced to zero by elementary transformation ?

I think Rank doesn't change because Rank of a matrix is the number of linearly independent row, which means the row which will be reduced to zero was never counted in finding the Rank of the matrix, so it will not change the Rank if it's reduced to zero.

But the solution in my reference book says Rank decreases if the elements of a row are reduced to zero by elementary transformation.

I am confused because I cant' find any logical explanation for it.

So once more I will type the question properly:

Q. By applying elementary Transformation to a matrix, it's rank:

(A) Does not change

(B) Decreases if the elements of a row are reduced to zero by elementary transformation.

asked Oct 08 '15 at 02:18

Pranav Peethambaran

Elementary row operations are reverrsiuble. So if the rank went down, that would mean the number of independent rows went down, but the latter number is just the dimension of the row space and so is constant under reversible linear operations, – coffeemath Oct 08 '15 at 02:22
You have got all your reasoning right. Look at it from equation view point. These manipulations do not change the solvability and the number of solutions. – Shailesh Oct 08 '15 at 02:32
Thanks. So the answer is (B) right? @coffeemath – Pranav Peethambaran Oct 08 '15 at 06:31
@PranavPeethambaran The answer is (A), Rank does not change. – Shailesh Oct 08 '15 at 06:35
@Shailesh: Ok. Thanks for correcting me! :) – Pranav Peethambaran Oct 08 '15 at 06:38

does the elementary transformation to a matrix change the rank?

0 Answers0