# Using elementary row and coloumn transformation compute rank of following matrices: ((25,31,17,43),(75,94,53,132),(75,94,54,134),(25,32,20,48)) ?

Jul 28, 2018

#### Answer:

The $R a n k \left(A\right) = 3$

#### Explanation:

The matrix is

$A = \left(\begin{matrix}25 & 31 & 17 & 43 \\ 75 & 94 & 53 & 132 \\ 75 & 94 & 54 & 134 \\ 25 & 32 & 20 & 48\end{matrix}\right)$

Perform the following operations :

$R 2 \leftarrow \left(R 2 - 3 R 1\right)$ ; $R 3 \leftarrow \left(R 3 - 3 R 1\right)$ ; $R 4 \leftarrow \left(R 4 - 3 R 1\right)$

$= \left(\begin{matrix}25 & 31 & 17 & 43 \\ 0 & 1 & 2 & 3 \\ 0 & 1 & 3 & 5 \\ 0 & 1 & 3 & 5\end{matrix}\right)$

$R 4 \leftarrow \left(R 4 - R 3\right)$ ; $R 3 \leftarrow \left(R 3 - R 2\right)$

$= \left(\begin{matrix}25 & 31 & 17 & 43 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0\end{matrix}\right)$

$R 1 \leftarrow \frac{R 1}{25}$

$= \left(\begin{matrix}1 & 1.24 & 0.68 & 1.72 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0\end{matrix}\right)$

Since there are $3$ non -zero rows, the

$R a n k \left(A\right) = 3$