Question

Find the rank of the matrix $\left[\begin{array}{cccc}0&1&1&-2\\4&0&2&5\\2&1&3&1\end{array}\right]$

Answer 1

Answer:

Rank of matrix=3

Step-by-step explanation:

The rank of the matrix is equal to the number of nonzero rows in the matrix after reducing it to the row echelon form using elementary transformations over the rows of the matrix.

so here

$\left[\begin{array}{cccc}0&1&1&-2\\4&0&2&5\\2&1&3&1\end{array}\right] \\\\\\R_{2}<->R_{1}\\\\ \left[\begin{array}{cccc}4&0&2&5\\0&1&1&-2\\2&1&3&1\end{array}\right] \\\\\\R_{3}->R_{3}-\frac{1}{2}R_{1}\\\\\left[\begin{array}{cccc}4&0&2&5\\0&1&1&-2\\0&1&2&\farc{-3}{2}\end{array}\right]\\\\\\R_{3}->R_{3}-R_{2}\\\\\left[\begin{array}{cccc}4&0&2&5\\0&1&1&-2\\0&0&1&\frac{1}{2} \end{array}\right]$

$Rank\left[\begin{array}{cccc}0&1&1&-2\\4&0&2&5\\2&1&3&1\end{array}\right] =Rank\left[\begin{array}{cccc}4&0&2&5\\0&1&1&-2\\0&0&1&\frac{1}{2} \end{array}\right]=3$