Question

Rank of matrix:

6 1 3 8
4 2 6 -1
10 3 9 7
16 4 12 15

Answer 1

Rank of matrix $\left[\begin{array}{cccc}6&1&3&8\\4&2&6&-1\\10&3&9&7\\16&4&12&15\end{array}\right]$ is 2.

Given:

• A 4×4 matrix.
• $\left[\begin{array}{cccc}6&1&3&8\\4&2&6&-1\\10&3&9&7\\16&4&12&15\end{array}\right]$

To find:

• Rank of matrix.

Solution:

Concept to be used:Convert the matrix in Row-echlon form and count the number of linearly independent rows.

Step 1:

Apply elementary row operations.

$R_1 = > \frac{R_1}{6}$

$\left[\begin{array}{cccc}1& \frac{1}{6} & \frac{1}{2} & \frac{4}{3} \\4&2&6&-1\\10&3&9&7\\16&4&12&15\end{array}\right]$

Step 2:

$R_2 = > R_2 - 4R_1$

$\left[\begin{array}{cccc}1& \frac{1}{6} & \frac{1}{2} & \frac{4}{3} \\ \\ 0& \frac{4}{3} &4& \frac{ - 19}{3} \\ \\ 10&3&9&7\\16&4&12&15\end{array}\right]$

Step 3:

$R_3 = > R_3 - 10R_1$

$\left[\begin{array}{cccc}1& \frac{1}{6} & \frac{1}{2} & \frac{4}{3} \\ \\ 0& \frac{4}{3} &4& \frac{ - 19}{3} \\ \\ 0& \frac{4}{3} &4& - \frac{19}{3} \\ \\ 16&4&12&15\end{array}\right]$

Step 4:

$R_4 = > R_4 - 16R_1$

$\left[\begin{array}{cccc}1& \frac{1}{6} & \frac{1}{2} & \frac{4}{3} \\ \\ 0& \frac{4}{3} &4& \frac{ - 19}{3} \\ \\ 0& \frac{4}{3} &4& - \frac{19}{3} \\ \\ 0& \frac{4}{3} &4& - \frac{19}{3} \\ \end{array}\right]$

Step 5:

$R_4 = > R_4 - R_2$

$\left[\begin{array}{cccc}1& \frac{1}{6} & \frac{1}{2} & \frac{4}{3} \\ \\ 0& \frac{4}{3} &4& \frac{ - 19}{3} \\ \\ 0& \frac{4}{3} &4& - \frac{19}{3} \\ \\ 0&0 &0& 0\\ \end{array}\right]$

Step 6:

$R_3 = > R_3- R_2$

$\left[\begin{array}{cccc}1& \frac{1}{6} & \frac{1}{2} & \frac{4}{3} \\ \\ 0& \frac{4}{3} &4& \frac{ - 19}{3} \\ \\ 0&0 &0& 0 \\ \\ 0&0 &0& 0\\ \end{array}\right]$

There are two linearly independent rows.

Thus,

Rank of matrix is 2.

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Learn more:

1) Find the rank of the matrix $\left[\begin{array}{ccc}1&1\\0&0\end{array}\right]$

https://brainly.in/question/7029426

2) Find the rank of the matrix $\left[\begin{array}{ccc}1&2&6\\2&4&3\end{array}\right]$

https://brainly.in/question/7029434

Answer 2

Answer:

Step-by-step explanation: