The rank of the matrix begin{bmatrix} -1 & 2 | vedclass

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(B) Let $A = \begin{bmatrix} -1 & 2 & 5 \ 2 & -4 & a - 4 \ 1 & -2 & a + 1 \end{bmatrix}$.Applying row operations $R_2 	o R_2 + 2R_1$ and $R_3 	o R

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$2$ if $a = 1$

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(B) Let $A = \begin{bmatrix} -1 & 2 & 5 \ 2 & -4 & a - 4 \ 1 & -2 & a + 1 \end{bmatrix}$.Applying row operations $R_2 	o R_2 + 2R_1$ and $R_3 	o R_3 + R_1$:$A \sim \begin{bmatrix} -1 & 2 & 5 \ 0 & 0 & a + 6 \ 0 & 0 & a + 6 \end{bmatrix}$.Applying $R_3 	o R_3 - R_2$:$A \sim \begin{bmatrix} -1 & 2 & 5 \ 0 & 0 & a + 6 \ 0 & 0 & 0 \end{bmatrix}$.If $a = -6$,the matrix becomes $\begin{bmatrix} -1 & 2 & 5 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix}$,so $ho(A) = 1$.If $a 
eq -6$,the second row is non-zero,so $ho(A) = 2$.Checking the options:For $a = 6$,$ho(A) = 2$ (Option $A$ is incorrect).For $a = 1$,$ho(A) = 2$ (Option $B$ is correct).For $a = 2$,$ho(A) = 2$ (Option $C$ is incorrect).Thus,the correct option is $B$.

The rank of the matrix $\begin{bmatrix} -1 & 2 & 5 \\ 2 & -4 & a - 4 \\ 1 & -2 & a + 1 \end{bmatrix}$ is

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