Question : Find row reduce form of the following matrix
\[M=\left(\begin{matrix}3&-1&2&4\\9&-7&1&-2\end{matrix}\right)\]
Solution :
Given matrix
\[M=\left(\begin{matrix}3&-1&2&4\\9&-7&1&-2\end{matrix}\right)\]
To find rank of M
Step 1:
Applying elementary row transformation \(R_2\rightarrow R_2-3R_1\)
\[M~\left(\begin{matrix}3&-1&2&4\\0&-4&-5&-14\\\end{matrix}\right)\]
Step 2 :
applying elementary row transformation \(R_1\rightarrow R_1-4R_2\)
\[M~\left(\begin{matrix}3&0&22&60\\0&-4&-5&-14\end{matrix}\right)\]
Step 3:
Applying elementary row transformation \(R_1\rightarrow\frac{R_1}{3}\)
\[M~\left(\begin{matrix}1&0&\frac{22}{3}&20\\0&-4&-5&-14\end{matrix}\right)\]
Step 4:
Applying elementary row transformation \(R_2\rightarrow\frac{R_2}{-4}\) \
\[M~\left(\begin{matrix}1&0&\frac{22}{3}&20\\0&1&\frac{5}{4}&\frac{7}{2}\end{matrix}\right)\]
Is the row reduce form of the given matrix.
Since there are two non zero rows in the row reduce form of M Hence rank of M is 2.
\[\rho(M) =2\]