When is the rank of a matrix equal to 1?

Answer 1

First you need to understand the definition of linear independence.

Two row vectors are independent if the first vector is NOT a multiple of the other.

For example: <1 0 1> and <0 1 0> are linearly independent. You cannot multiply one

of them by another to get the other.

When the Matrix is reduced to complete row-echelon form (nothing but zeros and ones).

Then each row vector are linearly independent: the one appears in a different position.

A good matrix has 1 along the main diagonal and zero everywhere else.

If so, the rank of the matrix is N, where n is the # of rows.

So if the rank of the matrix is 1, then the matrix has only 1 row vector that is linearly independent

from all the rest.

So the 1-D matrix looks like this [1]

The 2-D matrix looks like this

[1 0]

[0 0]

or

[0 1]

[0 0]

or

[1 1]

[ 0 0]

The 3-D matrix looks like this:

[1 0 0]

[ 0 0 0]

[0 0 0]

or

[0 1 0]

[0 0 0]

[0 0 0]

or

[0 0 1]

[0 1 0]

[ 0 0 1]

or

[ 1 1 0] and zeros everywhere else

or

[1 0 1] and zeros everywhere else

or

[0 1 1] and zeros everywhere else

or [1 1 1] and zeros everywhere else

etc etc etc

In contrast, the following matrices have rank = 2

[1 0]'

[0 1]

[0 1 1]

[1 0 0]

[0 0 0 ]

because 2 rows are linearly independent.

Do you understand?