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The rank of the matrix $\left[\begin{array}{ccc}1 & 0 & 2 \\ 0 & 1 & -2 \\ 1 & -1 & 4 \\ 2 & 2 & 8\end{array}\right]$ is
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Verified Answer
The correct answer is:
$3$
Let $A$
$$
\begin{aligned}
=\left[\begin{array}{ccc}
1 & 0 & 2 \\
0 & 1 & -2 \\
1 & -1 & 4 \\
2 & 2 & 8
\end{array}\right] & \sim\left[\begin{array}{ccc}
1 & 0 & 2 \\
0 & 1 & -2 \\
0 & -1 & 2 \\
0 & 2 & 4
\end{array}\right]\left[R_3 \rightarrow R_3-R_1 ; R_4 \rightarrow R_4-2 R_1\right] \\
& \sim\left[\begin{array}{ccc}
1 & 0 & 2 \\
0 & 1 & -2 \\
0 & 0 & 0 \\
0 & 0 & 8
\end{array}\right]\left[R_3 \rightarrow R_3+R_2 ; R_4 \rightarrow R_4-2 R_2\right]
\end{aligned}
$$
Number of non-zero rows is 3
$\therefore$ Rank of $A=3$
$$
\begin{aligned}
=\left[\begin{array}{ccc}
1 & 0 & 2 \\
0 & 1 & -2 \\
1 & -1 & 4 \\
2 & 2 & 8
\end{array}\right] & \sim\left[\begin{array}{ccc}
1 & 0 & 2 \\
0 & 1 & -2 \\
0 & -1 & 2 \\
0 & 2 & 4
\end{array}\right]\left[R_3 \rightarrow R_3-R_1 ; R_4 \rightarrow R_4-2 R_1\right] \\
& \sim\left[\begin{array}{ccc}
1 & 0 & 2 \\
0 & 1 & -2 \\
0 & 0 & 0 \\
0 & 0 & 8
\end{array}\right]\left[R_3 \rightarrow R_3+R_2 ; R_4 \rightarrow R_4-2 R_2\right]
\end{aligned}
$$
Number of non-zero rows is 3
$\therefore$ Rank of $A=3$
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