What is the rank of a matrix when the difference between its rows is same?

Question

I have this matrix: $$ \begin{pmatrix} 1& 3 &2\\ 2 &4& 3\\ 3& 5& 4 \end{pmatrix} $$ I noticed that `$R_2 - R_1 = R_3 - R_2$

How many rows are dependent in this case? and what will be the rank?

Answer 1

Hint:

You have show that $$ R_1-2R_2+R_3=0 $$

so the rows are linearly dependent.

but any couple of rows are linearly independent because , as you can easily see, there is not a real number $k$ such that $kR_1=R_2$ or $kR_1=R_3$ or $kR_3=R_2$ . So...

Answer 2

If you solve your equation for $R_2$ you get: $R_2 = \frac{1}{2}(R_1+R_3)$ so the matrix cannot have full rank. Gaussian elimination can be used to arrive at the rank. In this case you will see one row of zero's and the matrix has 2 pivots and so the rank is 2.