Solved find the rank of the following matrix

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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.

\[\rho(M) =2\]


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