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Problem 76: Find the rank of the following matrices by using Row reduction form:

Problem 76: Find the rank of the following matrices by using Row reduction form:

$(1).$ \begin{align*} \left[\begin{array}{rr} 2 & 1 \\ 1 & -3\\ \end{array}\right]\\ \end{align*} $(2).$ \begin{align*} \left[\begin{array}{rr} 2 & -4 \\ -4 & 8\\ \end{array}\right] \end{align*}

Solution:

$(1).$ \begin{align*} \left[\begin{array}{rr} 2 & 1 \\ 1 & -3\\ \end{array}\right] \xrightarrow{R_1 \leftrightarrow R_2} \left[\begin{array}{rr} 1 & -3\\ 2 & 1 \\ \end{array}\right] \end{align*} \[\xrightarrow{R_2\rightarrow R_2-2R_1} \left[\begin{array}{rr} 1 & -3\\ 0 & 7 \\ \end{array}\right] \xrightarrow{R_2\rightarrow \frac{1}{7}R_2} \left[\begin{array}{rr} 1 & -3\\ 0 & 1 \\ \end{array}\right]\]

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

$(2).$ \begin{align*} \left[\begin{array}{rr} 2 & -4 \\ -4 & 8\\ \end{array}\right] \xrightarrow{R_1\rightarrow \frac{1}{2}R_1} \left[\begin{array}{rr} 1 & -2 \\ -4 & 8\\ \end{array}\right] \end{align*} \[\xrightarrow{R_2\rightarrow R_2+4R_1} \left[\begin{array}{rr} 1 & -2\\ 0 & 0 \\ \end{array}\right]\]

Number of non-zero rows in RRF are 1.
Hence rank is 1.

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