The rank of the matrix -1252-4a-41-2a+1 is

A
1 if a = 6
B
2 if a = 1
C
3 if a = 2
D
1 if a = –6

Solution:

$A = [\begin{matrix} - 1 & 2 & 5 \\ 2 & - 4 & a + 4 \\ 1 & - 2 & a + 1 \end{matrix}] = [\begin{matrix} 0 & 0 & a + 6 \\ 0 & 0 & - a + 6 \\ 1 & - 2 & a + 1 \end{matrix}]$

(operating $R_{1} \to R_{1} + R_{3} a n d R_{2} \to R_{2} - 2 R_{3}$ )

$= [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & - a - 6 \\ 1 & - 2 & a + 1 \end{matrix}] (O p e r a t i n g R_{1} \to R_{1} + R_{2})$

$W h e n a = - 6, A = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & - 2 & - 5 \end{matrix}], ∴ ρ (A) = 1$

$W h e n a = 6, A = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & - 12 \\ 1 & - 2 & - 5 \end{matrix}], ∴ ρ (A) = 2$

$W h e n a = 1, A = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & - 7 \\ 1 & - 2 & - 2 \end{matrix}], ∴ ρ (A) = 2$

$W h e n a = 2, A = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & - 8 \\ 1 & - 2 & 3 \end{matrix}], ∴ ρ (A) = 2$
The correct option is (B) and (D)

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

Solution:

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