# Rank of Matrix ??

• Oct 8th 2008, 03:27 AM
Sameera
Rank of Matrix ??
Can someone please explain "rank of matix" through example ????

THANKS.....
• Oct 8th 2008, 08:58 AM
ThePerfectHacker
Quote:

Originally Posted by Sameera
Can someone please explain "rank of matix" through example ????

THANKS.....

The rank of a matrix is a dimension of the spam of the its rows.

For example, consider:
$A=\begin{bmatrix} 1 & 2 & 3 \\ 1 & 2 & 4 \\ 2&4&6 \end{bmatrix}$

The rows are:
1) $\bold{r}_1 = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$
2) $\bold{r}_2 = \begin{bmatrix} 1 & 2 & 4 \end{bmatrix}$
3) $\bold{r}_3 = \begin{bmatrix} 2 & 4 & 6 \end{bmatrix}$

We want to find the dimension of $\text{spam}\{ \bold{r}_1, \bold{r}_2, \bold{r}_3 \}$.

Note that $\bold{r}_3 = 2\bold{r}_1$ therefore $\text{spam}\{ \bold{r}_1,\bold{r}_2,\bold{r}_3 \} = \text{spam}\{ \bold{r}_1, \bold{r}_2\}$.
And $\bold{r}_1,\bold{r}_2$ are linearly independent since they are not proportional.
Therefore, the dimension of $\text{spam}\{\bold{r}_1,\bold{r}_2\}$ is $2$.

Therefore $\text{rank}(A) = 2$