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Matrices and Determinants question

2007 · Shift 0 · Q51

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JEE Main
Mathematics
Matrices and Determinants
2007 · Shift 0 · Q51

Matrices and Determinants question

2007 · Shift 0 · Q51

JEE MainMathematicsMatrices and DeterminantsMCQ+4 / −1

If

D = \left| {\begin{matrix} 1 & 1 & 1 \\ 1 & {1 + x} & 1 \\ 1 & 1 & {1 + y} \\ \end{matrix} } \right|

for

x e 0,y e 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 $y$

View written solutionFree

Correct answer: D

Given determinant

We need to evaluate

D=\begin{vmatrix} 1&1&1\\ 1&1+x&1\\ 1&1&1+y \end{vmatrix}, \qquad x\ne 0,\ y\ne 0.

We must determine whether $D$ is divisible by $x$ , by $y$ , by both, or by neither.

Use row operations to simplify

Apply row operations that do not change the determinant:

$R_2 \to R_2 - R_1$
$R_3 \to R_3 - R_1$

Then

D=\begin{vmatrix} 1&1&1\\ 0&x&0\\ 0&0&y \end{vmatrix}.

Evaluate the determinant

Now the matrix is upper triangular, so the determinant is the product of diagonal entries:

D=1\cdot x\cdot y = xy.

Check divisibility

Since

D=xy,

it is clearly divisible by $x$ and also divisible by $y$ .

So the correct option is:

\boxed{\text{D: divisible by both } x \text{ and } y}

Comparison with stored answer

Stored correct answer: D

Our derived answer: D

Hence, the derived answer agrees with the stored answer.

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