(A) Let $\Delta = \left|\begin{array}{ccc}1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y\end{array}\right|$.Applying row operations $R_{2} \rightarrow R_{2}-

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(A) Let $\Delta = \left|\begin{array}{ccc}1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y\end{array}\right|$.Applying row operations $R_{2} \rightarrow R_{2}-R_{1}$ and $R_{3} \rightarrow R_{3}-R_{1}$,we get:$\Delta = \left|\begin{array}{ccc}1 & x & y \\ 0 & y & 0 \\ 0 & 0 & x\end{array}\right|$.Expanding the determinant along the first column $(C_{1})$:$\Delta = 1 \cdot \left|\begin{array}{cc}y & 0 \\ 0 & x\end{array}\right| - 0 + 0$$\Delta = 1 \cdot (yx - 0) = xy$.Thus,the value of the determinant is $xy$.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Expansion of determinants, Solution of equation in the form of determinants and area of triangle and Equation of Line

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Evaluate $\left|\begin{array}{ccc}1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y\end{array}\right|$

A
$xy$
B
$x^2y$
C
$x^2y^2$
D
$xy^2$

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3 and 4 .Determinants and Matrices

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Expansion of determinants, Solution of equation in the form of determinants and area of triangle and Equation of Line

If $a+x=b+y=c+z+1,$ where $a, b, c, x, y, z$ are non-zero distinct real numbers,then $\left|\begin{array}{lll}x & a+y & x+a \\ y & b+y & y+b \\ z & c+y & z+c\end{array}\right|$ is equal to

DifficultJEE MAIN 2020

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The value of the determinant $\left| \begin{array}{ccc} 31 & 37 & 92 \\ 31 & 58 & 71 \\ 31 & 105 & 24 \end{array} \right|$ is

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$\left| {\begin{array}{ccc} 1 + i & 1 - i & i \\ 1 - i & i & 1 + i \\ i & 1 + i & 1 - i \end{array}} \right| = $

Medium

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If $1, \omega, \omega^2$ are the cube roots of unity,then $\Delta = \begin{vmatrix} 1 & \omega^n & \omega^{2n} \\ \omega^n & \omega^{2n} & 1 \\ \omega^{2n} & 1 & \omega^n \end{vmatrix} = $

MediumAIEEE 2003

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The roots of the equation $\left| \begin{matrix} 1+x & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+x \end{matrix} \right| = 0$ are

Easy

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Evaluate $\left|\begin{array}{ccc}1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y\end{array}\right|$

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If $a+x=b+y=c+z+1,$ where $a, b, c, x, y, z$ are non-zero distinct real numbers,then $\left|\begin{array}{lll}x & a+y & x+a \\ y & b+y & y+b \\ z & c+y & z+c\end{array}\right|$ is equal to

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