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If $A=\left[\begin{array}{ccccc}0 & 1 & 2 & 3 & 4 \\ 0 & 2 & 4 & 8 & 12 \\ 0 & 0 & 0 & 4 & 8\end{array}\right]$, then the rank of $A$ is
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The correct answer is:
$2$
$A=\left[\begin{array}{lllll}0 & 1 & 2 & 3 & 4 \\ 0 & 2 & 4 & 8 & 12 \\ 0 & 0 & 0 & 4 & 8\end{array}\right]$
$\mathrm{R}_2 \rightarrow \mathrm{R}_2-2 \mathrm{R}^1$
$\begin{aligned} & \sim\left[\begin{array}{lllll}0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 2 & 4 \\ 0 & 0 & 0 & 4 & 8\end{array}\right] \\ & \because\left|\begin{array}{cc}3 & 4 \\ 2 & 4\end{array}\right|=12-8=4 \uparrow 0\end{aligned}$
$\therefore$ Rank $=2$
$\mathrm{R}_2 \rightarrow \mathrm{R}_2-2 \mathrm{R}^1$
$\begin{aligned} & \sim\left[\begin{array}{lllll}0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 2 & 4 \\ 0 & 0 & 0 & 4 & 8\end{array}\right] \\ & \because\left|\begin{array}{cc}3 & 4 \\ 2 & 4\end{array}\right|=12-8=4 \uparrow 0\end{aligned}$
$\therefore$ Rank $=2$
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