Q45 / 160 · AP EAMCET 2023 (19 May Shift 1) · Mathematics

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

  • A
    $2$
  • B
    $1$
  • Correct answer
    C
    $3$
  • D
    $4$

Solution

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 $A=3$

Hence, the correct answer is option: 'C'.

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