Evaluate left|begin{array}{ccc}x & y & x+y y | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $\Delta = \left|\begin{array}{ccc}x & y & x+y \ y & x+y & x \ x+y & x & y\end{array}ight|$.Applying the row operation $R_{1} ightarrow R

Vedclass · Accepted Answer

$-2(x^{3}+y^{3})$

Vedclass · Answer

(A) Let $\Delta = \left|\begin{array}{ccc}x & y & x+y \ y & x+y & x \ x+y & x & y\end{array}ight|$.Applying the row operation $R_{1} ightarrow R_{1} + R_{2} + R_{3}$,we get:$\Delta = \left|\begin{array}{ccc}2(x+y) & 2(x+y) & 2(x+y) \ y & x+y & x \ x+y & x & y\end{array}ight|$.Taking $2(x+y)$ common from $R_{1}$:$\Delta = 2(x+y) \left|\begin{array}{ccc}1 & 1 & 1 \ y & x+y & x \ x+y & x & y\end{array}ight|$.Applying column operations $C_{2} ightarrow C_{2} - C_{1}$ and $C_{3} ightarrow C_{3} - C_{1}$:$\Delta = 2(x+y) \left|\begin{array}{ccc}1 & 0 & 0 \ y & x & x-y \ x+y & -y & -x\end{array}ight|$.Expanding along $R_{1}$:$\Delta = 2(x+y) [1 \cdot (x(-x) - (-y)(x-y))]$.$\Delta = 2(x+y) [-x^{2} + y(x-y)]$.$\Delta = 2(x+y) [-x^{2} + xy - y^{2}]$.$\Delta = -2(x+y)(x^{2} - xy + y^{2})$.Using the identity $a^{3} + b^{3} = (a+b)(a^{2} - ab + b^{2})$:$\Delta = -2(x^{3} + y^{3})$.

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

Similar Questions

If $\left[\begin{array}{rrr}1 & -1 & x \\ 1 & x & 1 \\ x & -1 & 1\end{array}\right]$ has no inverse, then the real value of $x$ is

Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x+y+z=1$,$2x+Ny+2z=2$,and $3x+3y+Nz=3$ has a unique solution is $\frac{k}{6}$,then the sum of the value of $k$ and all possible values of $N$ is:

If $\omega$ is the cube root of unity,then $\left| \begin{array}{ccc} 1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \end{array} \right| = $

The value of $\det A$, where $A = \begin{bmatrix} 1 & \cos \theta & 0 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{bmatrix}$, lies

If $\left| \begin{array}{ccc} 3x - 8 & 3 & 3 \\ 3 & 3x - 8 & 3 \\ 3 & 3 & 3x - 8 \end{array} \right| = 0$,then the values of $x$ are

Vedclass Test Series

Exam Paper Generator

Online Exam Module