Question

Can you tell me how to calculate R for below Matrix ?

x1 =9 2 6 5 8

x2= 12 8 6 4 10

x3= 3 4 0 2 1

Answer 1

The rank of the matrix is 3.

Explanation:

Consider the prided information.

`x=\begin{bmatrix}x_1\\ x_2\\ x_3\end{bmatrix}=\begin{bmatrix}9&2&6&5&8\\ 12&8&6&4&10\\ 3&4&0&2&1\end{bmatrix}`

Reduce the matrix in row echelon form as shown:

`R_1\:\leftrightarrow \:R_2\begin{bmatrix}12&8&6&4&10\\ 9&2&6&5&8\\ 3&4&0&2&1\end{bmatrix}\\`

`R_2\:\leftarrow \:R_2-(3)/(4)\cdot \:R_1\ and\ R_3\:\leftarrow \:R_3-(1)/(4)\cdot \:R_1\\\\\begin{bmatrix}12&8&6&4&10\\ 0&-4&(3)/(2)&2&(1)/(2)\\ 0&2&-(3)/(2)&1&-(3)/(2)\end{bmatrix}`

`R_3\:\leftarrow \:R_3+(1)/(2)\cdot \:R_2\\\begin{bmatrix}12&8&6&4&10\\ 0&-4&(3)/(2)&2&(1)/(2)\\ 0&0&-(3)/(4)&2&-(5)/(4)\end{bmatrix}`

`R_2\:\leftarrow \:R_2-(3)/(2)\cdot \:R_3\ and\ R_1\:\leftarrow \:R_1-6\cdot \:R_3\\\begin{bmatrix}12&8&0&20&0\\ 0&-4&0&6&-2\\ 0&0&1&-(8)/(3)&(5)/(3)\end{bmatrix}\\R_2\:\leftarrow \:-(1)/(4)\cdot \:R_2\ , \ R_1\:\leftarrow \:R_1-8\cdot \:R_2\ and\ \:R_1\:\leftarrow (1)/(12)\cdot \:R_1\\\begin{bmatrix}1&0&0&(8)/(3)&-(1)/(3)\\ 0&1&0&-(3)/(2)&(1)/(2)\\ 0&0&1&-(8)/(3)&(5)/(3)\end{bmatrix}`

The rank of a matrix is the number of non zeros rows.

Thus, the rank of the matrix is 3.