Find the rank of this 3x3 matrix

[PainterGuy]

Homework Statement

Where am I going wrong with finding the rank?

Relevant Equations

Please check my attempt.

Hi,

I was trying to find the rank of following matrix.

I formed the following system and it seems like all three columns are linearly independent and hence the rank is 3. But the answer says the rank is '2'. Where am I going wrong? Thanks, in advance!

 

[Office_Shredder]

Why do you think $k_1=0$ in your equation?

Equivalently, it is very obvious (after some experience) that the three columns are not linearly independent.

 

[Delta2]

One of the columns is the zero column (all entries zero). Doesn't that tell you something? Can the zero vector be linearly independent with any other vector(s)?

 

[nuuskur]

There exists a non-zero subdeterminant (or minor) of order two and the determinant of the entire thing is 0. Equivalently, the rank is two. The second equality adds no information. The rank can be at most two.

 

[epenguin]

Reason you can't see it is probably you've done and got used to, routinised, other more or less difficult exercises but never met anything this trivial!

Just check definition of linear dependence "... said to be linearly dependent, if there exist scalars

not all zero, such that

where

denotes the zero vector."

 

[PainterGuy]

Thank you, everyone!

Reason you can't see it is probably you've done and got used to, routinised, other more or less difficult exercises but never met anything this trivial!

Or, I'm just silly! :)

Source: Linear Algebra, 6th ed. by Seymour

 

[PeroK]

Thank you, everyone!
Or, I'm just silly! :)

One of the first things you should learn about the rank of a matrix is to discard any rows (or colums) that are zero.