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Problem 77: Find the rank of the following matrices by using Row reduction form

Problem 77: Find the rank of the following matrices by using Row reduction form:

$(1).$ \begin{align*} \left[\begin{array}{rrr} 2 & 1& 4 \\ 2 & -3& 4 \\ 3 & -2& 6 \\ \end{array}\right]\\ \end{align*} $(2).$ \begin{align*} \left[\begin{array}{rrr} 0 & 1& 3 \\ 0 & 1& 4 \\ 0 & 3& 5 \\ \end{array}\right]\\ \end{align*}

Solution:

$(1).$ \begin{align*} \left[\begin{array}{rrr} 2 & 1& 4 \\ 2 & -3& 4 \\ 3 & -2& 6 \\ \end{array}\right] \xrightarrow{R_1\rightarrow \frac{1}{2}R_1} \left[\begin{array}{rrr} 1 & \frac{1}{2}& 2 \\ 2 & -3& 4 \\ 3 & -2& 6 \\ \end{array}\right] \end{align*} \[\xrightarrow{R_2\rightarrow R_2-2R_1,R_3\rightarrow R_3-3R_1} \left[\begin{array}{rrr} 1 & \frac{1}{2}& 2 \\ 0 & -4& 0 \\ 0 & -\frac{7}{2}& 0 \\ \end{array}\right] \xrightarrow{R_2\rightarrow -\frac{1}{4}R_2} \left[\begin{array}{rrr} 1 & \frac{1}{2}& 2 \\ 0 & 1& 0 \\ 0 & -\frac{7}{2}& 0 \\ \end{array}\right] \xrightarrow{R_3\rightarrow R_3+\frac{2}{7}R_1} \left[\begin{array}{rrr} 1 & \frac{1}{2}& 2 \\ 0 & 1& 0 \\ 0 & 0& 0 \\ \end{array}\right]\]

Number of non-zero rows in RRF are 2.
Hence rank is 2.

$(2).$ \begin{align*} \left[\begin{array}{rrr} 0 & 1& 3 \\ 0 & 1& 4 \\ 0 & 3& 5 \\ \end{array}\right] \xrightarrow{R_2\rightarrow R_2-R_1,R_3\rightarrow R_3-3R_1} \left[\begin{array}{rrr} 0 & 1& 3 \\ 0 & 0& 1 \\ 0 & 0& -4 \\ \end{array}\right] \end{align*} \[\xrightarrow{R_3\rightarrow R_2+4R_2} \left[\begin{array}{rrr} 0 & 1& 3 \\ 0 & 0& 1 \\ 0 & 0& 0 \\ \end{array}\right]\]

Number of non-zero rows in RRF are 2.
Hence rank is 2.

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