# Rank of Matrix ??

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• Oct 8th 2008, 02:27 AM
Sameera
Rank of Matrix ??
Can someone please explain "rank of matix" through example ????

THANKS.....
• Oct 8th 2008, 07:58 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:
$\displaystyle A=\begin{bmatrix} 1 & 2 & 3 \\ 1 & 2 & 4 \\ 2&4&6 \end{bmatrix}$

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

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

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

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