If the system
$2 x - y + 3 z = 2$
$x + y + 2 z = 2$
$5 x - y + a z = b$
has infinitely many solutions, then the values of a and b respectively, are

- 8 and 6

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8 and 6

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- 8 and - 6

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8 and - 6

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Solution

The correct option is B 8 and 6
$[\begin{matrix} A & : & B \end{matrix}] = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 2 & - 1 & 3 & ⋮ & 2 1 & 1 & 2 & ⋮ & 2 5 & - 1 & a & ⋮ & b \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$R_{1} \leftrightarrow R_{2} \sim ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 1 & 3 & ⋮ & 2 2 & - 1 & 3 & ⋮ & 2 5 & - 1 & a & ⋮ & b \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$R_{2} \to R_{2} - 2 R_{1} a n d R_{3} \to R_{3} - 5 R_{1}$

$[\begin{matrix} A : B \end{matrix}] \sim ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 1 & 2 & ⋮ & 2 0 & - 3 & - 1 & ⋮ & - 2 0 & - 6 & a - 10 & ⋮ & b - 10 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$R_{3} \to R_{3} - 2 R_{2}$ ,

$[\begin{matrix} A : B \end{matrix}] \sim ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 1 & 2 & ⋮ & 2 0 & - 3 & - 1 & ⋮ & - 2 0 & 0 & a - 8 & ⋮ & b - 6 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

Which is in Echelon form.
Now for infinite solutions,
$ρ (A) = ρ (A : B) < 3$
$\Rightarrow a - 8 = 0, b - 6 = 0$
$\Rightarrow a = 8, b = 6$

Suggest Corrections

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