# Rank of a matrix

• Apr 1st 2010, 05:17 PM
mybrohshi5
Rank of a matrix
Find the rank of the matrix

A = $\displaystyle \begin{bmatrix}0&6\\0&-2\\0&5\end{bmatrix}$

I know that rank is just the number of pivots a matrix has when in reduced row echelon form, but this one is confusing to me.

RREF A = $\displaystyle \begin{bmatrix}0&1\\0&0\\0&0\end{bmatrix}$

I thought this would have a rank of 0 because there are no pivots but i was wrong and it has a rank of 1.

Why is this?

Thanks for any help :)
• Apr 1st 2010, 05:32 PM
harish21
Quote:

Originally Posted by mybrohshi5
Find the rank of the matrix

A = $\displaystyle \begin{bmatrix}0&6\\0&-2\\0&5\end{bmatrix}$

I know that rank is just the number of pivots a matrix has when in reduced row echelon form, but this one is confusing to me.

RREF A = $\displaystyle \begin{bmatrix}0&1\\0&0\\0&0\end{bmatrix}$

I thought this would have a rank of 0 because there are no pivots but i was wrong and it has a rank of 1.

Why is this?

Thanks for any help :)

The matrix RREF A has one "non-zero" row. that means the matrix A has one independent row vector. So its rank is 1.
• Apr 1st 2010, 05:37 PM
mybrohshi5
So finding rank is just the number of rows that has at least one non-zero entry in it?
• Apr 1st 2010, 05:48 PM
harish21
Quote:

Originally Posted by mybrohshi5
So finding rank is just the number of rows that has at least one non-zero entry in it?

Yes, this is the way that I learnt to find the rank of a matrix. I would also suggest referring to linear independence

Actually, the NUMBER of rows (in the RREF matrix) that are NON-ZERO is the RANK of the matrix. Here your matrix has one $\displaystyle non-zero$ row, your rank is 1.
• Apr 1st 2010, 05:50 PM
mybrohshi5
Thanks that clears things up for me :)