Rank of a Matrix - Explained by an Expert

[NonameNoface]

Well this is eating my head ! or am I plain stupid ? ...

The rank of a Matrix , is determined by the number of independent rows or columns ..Fine .. here's a matrix ..

A = 2 4 1 3
-1 -2 1 0
0 0 2 2
3 6 2 5

Apparently " the second column is twice the first column, and that the fourth column equals the sum of the first and the third. The first and the third columns are linearly independent, so the rank of A is two "

here is the part where I am getting confused .. what if I consider the rows ..I see that all rows are linearly independent ... If somebody coudl explain this to me please ...

Thanks a lot
Regards
Mano

 

[kthouz]

don't matter about those columns or rows. i think that you should only try to transorm that matrix into an echelon one and then you'll see yourself clearly independant rows which define the rank of that matrix

 

[Parthalan]

The row rank of a matrix is the number of linearly independent rows, and similarly, the column rank is the number of linearly independent columns. Most importantly, note that the two are always equal!

With
$$A = \left [ \begin{array}{cccc} 2 & 4 & 1 & 3 \\ -1 & -2 & 1 & 0 \\ 0 & 0 & 2 & 2 \\ 3 & 6 & 2 & 5 \end{array} \right ]$$,
and since we know the column rank of this matrix is 2, can you now find a way to express any 2 of the rows as a linear combination of the others?

 

[radou]

I see that all rows are linearly independent

Write down, don't just look.

 

[madhusudan]

Well this is eating my head ! or am I plain stupid ? ...

The rank of a Matrix , is determined by the number of independent rows or columns ..Fine .. here's a matrix ..

A = 2 4 1 3
-1 -2 1 0
0 0 2 2
3 6 2 5

Apparently " the second column is twice the first column, and that the fourth column equals the sum of the first and the third. The first and the third columns are linearly independent, so the rank of A is two "

here is the part where I am getting confused .. what if I consider the rows ..I see that all rows are linearly independent ... If somebody coudl explain this to me please ...

Thanks a lot
Regards
Mano

hey ... the rank of the matrix is 3.do the following operations to get echelon form...
r1<->r2;
r1<->-r1;
r2-->r2-2r1;
r4-->3r1-r4;
r4-->r3+r2+r4;

 

[madhusudan]

hey ... the rank of the matrix is 2.do the following operations to get echelon form...
r1<->r2;
r1<->-r1;
r2-->r2-2r1;
r4-->3r1-r4;
r4-->r3+r2+r4;

can you get it

 

[matt grime]

I see that all rows are linearly independent

Then perhaps a trip to the mathematical optician is in order. I can see that if I add the first row and twice the second row that...