# Find the rank of this 3x3 matrix

• PainterGuy
You are keeping a row of zeros, which should not be done. In summary, the conversation is discussing the rank of a matrix and the confusion over the answer being '2' when the columns are seemingly linearly independent. It is explained that the rank is actually '2' because one of the columns is the zero column, making the columns linearly dependent. The conversation also mentions the definition of linear dependence and the importance of discarding any rows or columns that are zero when determining the rank of a 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!

Last edited by a moderator:
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.

PeroK and PainterGuy
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)?

PainterGuy
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.

PainterGuy and Delta2
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!

epenguin said:
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

Delta2
PainterGuy said:
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.

PainterGuy

## What is a 3x3 matrix?

A 3x3 matrix is a rectangular array of numbers or variables arranged in three rows and three columns. It is commonly used in mathematics, physics, and engineering to represent linear equations and transformations.

## How do you find the rank of a 3x3 matrix?

The rank of a 3x3 matrix can be found by performing row operations, such as addition, subtraction, and multiplication, to reduce the matrix to its row echelon form. The number of non-zero rows in the row echelon form is the rank of the matrix.

## Why is finding the rank of a 3x3 matrix important?

The rank of a 3x3 matrix is important because it provides information about the linear independence of the rows or columns of the matrix. It also helps in solving systems of linear equations and determining the dimension of the vector space spanned by the matrix.

## What is the maximum rank of a 3x3 matrix?

The maximum rank of a 3x3 matrix is 3. This means that the matrix has three linearly independent rows or columns, and its row echelon form is a diagonal matrix with three non-zero entries.

## Can a 3x3 matrix have a rank of 0?

No, a 3x3 matrix cannot have a rank of 0. This is because a matrix with a rank of 0 would have all zero rows, which is not possible for a 3x3 matrix. The minimum rank of a 3x3 matrix is 1, meaning that it has at least one non-zero row or column.

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