If the system,

2x - y + 3z = 2

x + y + 2z = 2

5x - y + az = 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
Consider, $(A : B) = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 2 & | & 2 2 & - 1 & 3 & | & 2 5 & - 1 & a & | & b \end{matrix} ⎤ ⎥ ⎦$

Applying, $R_{2} \to R_{2} - 2 R_{1}, R_{3} \to R_{3} - 5 R_{1}$

$= ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 2 & | & 2 0 & - 3 & - 1 & | & - 2 0 & - 6 & a - 10 & | & b - 10 \end{matrix} ⎤ ⎥ ⎦$

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

$= ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 2 & | & 2 0 & - 3 & - 1 & | & - 2 0 & 0 & a - 8 & | & b - 6 \end{matrix} ⎤ ⎥ ⎦$

a = 8 and b = 6

$∴$ Infinite many solution.

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