MathematicsIf x,y,z are integers in A.P lying between 1 and 9 and x51, y41 and z31 are three digit number then the value of543x51y41z31xyz is

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

A
$x + y + z$
B
$x - y - z$
C
0
D
$x + 2 y + z$

Solution:

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

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