# Thread: Rank of Matrix ??

1. ## Rank of Matrix ??

Can someone please explain "rank of matix" through example ????

THANKS.....

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