If D = left| {,begin{array}{*{20}{c}}1&1&11&{1 + x}&11&1&{1 + y}end{array},}

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3 and 4 .Determinants and Matrices

medium

If $D = \left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{1 + x}&1\\1&1&{1 + y}\end{array}\,} \right|$ for $x \ne 0,y \ne 0$ then $D$ is

A

divisible by $x$ but not $y$

B

divisible by $y$ but not $x$

C

divisible by neither $x$ nor $y$

D

divisible by both $x$ and $y$

(AIEEE-2007)

Solution

Given, $D=\left|\begin{array}{ccc}{1} & {1} & {1} \\ {1} & {1+x} & {1} \\ {1} & {1} & {1+y}\end{array}\right|$

Apply $\quad R^{2} \rightarrow R_{2}-R_{1}$

and $\quad R \rightarrow R_{3}-R_{1}$

$\because \quad D=\left|\begin{array}{lll}{1} & {1} & {1} \\ {0} & {x} & {0} \\ {0} & {0} & {y}\end{array}\right|=x y$

Hence, $D$ is divisible by both $x$ and $y$

Standard 12

Mathematics

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