- #1

mr_coffee

- 1,629

- 1

I have the following:

2 3 -2

2 6 0

-4 0 0

Rank = 3;

0 2 0 0

0 0 0 -4

0 0 0 0

9 0 0 0

rank = 3;

1 2

6 -3

Rank = 2;

I don't get it! any help would be great!

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- Thread starter mr_coffee
- Start date

- #1

mr_coffee

- 1,629

- 1

I have the following:

2 3 -2

2 6 0

-4 0 0

Rank = 3;

0 2 0 0

0 0 0 -4

0 0 0 0

9 0 0 0

rank = 3;

1 2

6 -3

Rank = 2;

I don't get it! any help would be great!

- #2

Spectre5

- 182

- 0

The rank of a matrix is the number of linearly independent rows in the matrix.

You can find the rank by performing Gaussian elimination. The rank will then be the number of non-zero rows in the resulting matrix.

- #3

FrogPad

- 810

- 0

1) Let [tex] A [/tex] be a [tex] m\times n[/tex] matrix.

2) Perform gaussian-elimination on [tex] A [/tex]

3) Count the number of non zero columns. This number is the rank of the matrix.

We'll start with your first example.

1) [tex] A = \left[ \begin{array}{ccc}2 & 3 & -2 \\2 & 6 & 0 \\-4 & 0 & 0 \\\end{array}\right] [/tex]

2) Perform the gaussian-elimination on [tex] A [/tex]

[tex] A' = \left[ \begin{array}{ccc}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\\end{array}\right] [/tex]

3) Now we count the number of non-zero columns:

[tex] A' = \left[ \begin{array}{ccc}(1) & 0 & 0 \\0 & (1) & 0 \\0 & 0 & (1) \\\end{array}\right] [/tex]

Rank = 3

----

Now, what does the rank mean? Well the rank of a matrix let's you know the number of columns that cannot be written as a linear combination of each other.

If you consider [tex] A' [/tex] to be a matrix of vectors, we would have the following column vectors.

[tex] \vec{V_1} = (1,0,0) [/tex]

[tex] \vec{V_2} = (0,1,0) [/tex]

[tex] \vec{V_3} = (0,0,1) [/tex]

These vectors are ALL linearly independent of each other. Which means that the column vectors in [tex] A [/tex] are all linearly independent of each other.

- #4

amcavoy

- 665

- 0

- #5

mr_coffee

- 1,629

- 1

Awesome! thanks a lot guys! good explanation!

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