The rank of the matrix $A=\left[\begin{array}{lll}2 & 1 & 2 \ 1 & 0 & 1 \ 4 & 1 & 4\end{array} ight]$ is

Question

The rank of the matrix $A=\left[\begin{array}{lll}2 & 1 & 2 \ 1 & 0 & 1 \ 4 & 1 & 4\end{array}ight]$ is

Aicharya · Accepted Answer

Convert matrix into row echelon form. $ \begin{aligned} & {\left[\begin{array}{lll} 2 & 1 & 2 \ 1 & 0 & 1 \ 4 & 1 & 4 \end{array}ight] \xrightarrow{R_1 ightarrow R_1-R_2}\left[\begin{array}{lll} 1 & 1 & 1 \ 1 & 0 & 1 \ 4 & 1 & 4 \end{array}ight]} \ & {\left[\begin{array}{lll} 1 & 1 & 1 \ 1 & 0 & 1 \ 4 & 1 & 4 \end{array}ight] \underset{\substack{R_3 ightarrow R_3-4 R_1}}{R_2 ightarrow R_2-R_1}\left[\begin{array}{ccc} 1 & 1 & 1 \ 0 & -1 & 0 \ 0 & -3 & 0 \end{array}ight]} \ & {\left[\begin{array}{lll} 1 & 1 & 1 \ 0 & -1 & 0 \ 0 & -3 & 0 \end{array}ight] \xrightarrow[R_3 ightarrow\left(-\frac{1}{3}ight) R_3]{\stackrel{R_2 ightarrow(-1) R_2}{\longrightarrow}}\left[\begin{array}{lll} 1 & 1 & 1 \ 0 & 1 & 0 \ 0 & 3 & 0 \end{array}ight]} \ & {\left[\begin{array}{lll} 1 & 1 & 1 \ 0 & 1 & 0 \ 0 & 3 & 0 \end{array}ight] \xrightarrow{R_3 ightarrow R_3-R_2}\left[\begin{array}{lll} 1 & 1 & 1 \ 0 & 1 & 0 \ 0 & 0 & 0 \end{array}ight]} \end{aligned} $ Number of non-zero rows is 2 . $	herefore$ Rank of matrix is 2

The rank of the matrix $A=\left[\begin{array}{lll}2 & 1 & 2 \\ 1 & 0 & 1 \\ 4 & 1 & 4\end{array}\right]$ is

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