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If $x, y, z$ are integers in A.P lying between 1 and 9 and $x 51, y 41$ and $z 31$ are three digit number then the value of
$|\begin{matrix} 5 & 4 & 3 \\ x 51 & y 41 & z 31 \\ x & y & z \end{matrix}|$ is

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a

x + y + z

b

x - y - z

c

0

d

x + 2 y + z

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detailed solution

Correct option is C

$|\begin{matrix} 5 & 4 & 3 \\ x 51 & y 41 & z 31 \\ x & y & z \end{matrix}| R_{2} \to R_{2} - 100 R_{3} - 10 R_{1} {|\begin{matrix} 5 & 4 & 3 \\ 1 & 1 & 1 \\ x & y & z \end{matrix}|}_{C_{1} \to C_{1} + C_{3} - 2 C_{2}} \Rightarrow |\begin{matrix} 0 & 4 & 3 \\ 0 & 1 & 1 \\ 0 & y & z \end{matrix}| = 0 (x + z = 2 y)$

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