Using properties of determinants, prove the following: x&x + y &x… - Vidyadip

Question

Using properties of determinants, prove the following: $ \begin{bmatrix} \text{ x}&\text{x + y }&\text{x} + 2\text{y}\\ \text{x} + 2\text{y} & \text{x}& \text{x + y }\\\text{x + y}&\text{x} + 2\text{y}& \text{x} \end{bmatrix} = 9\text{y}^{2}(\text{x} + \text{y}). $

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Answer

L.H.S.$ =\begin{bmatrix} \text{ x}&\text{x + y }&\text{x} + 2\text{y}\\ \text{x} + 2\text{y} & \text{x}& \text{x + y }\\\text{x + y}&\text{x} + 2\text{y}& \text{x} \end{bmatrix} $
$ =\begin{bmatrix} 3(\text{x + y})&3(\text{x + y})&3(\text{x + y})\\ \text{x} + 2\text{y}&\text{x}&\text{x + y}\\ \text{x + y}&\text{x} + 2\text{y}&\text{x} \end{bmatrix}[\text{Applying R}_{1} = \text{R}_{1} + \text{R}_{2}+ \text{R}_{3}] $
$ =3(\text{x + y })\begin{bmatrix} 1 &1&1\\ \text{x} + 2\text{y}&\text{x}&\text{x + y}\\ \text{x + y}&\text{x} + 2\text{y}&\text{x} \end{bmatrix}\text{[ Taking 3 (x + y) common from R}_{1}]$
$ =3(\text{x + y })\begin{bmatrix} 0 &0&1\\ \text{y}&-\text{y}&\text{x + y} \\ \text{ y} & 2\text{y} &\text{x} \end{bmatrix}\text{[Applying C}_{1}\rightarrow\text{C}_{1} - \text{C}_{3} , \text{C}_{2}\rightarrow\text{C}_{2} - \text{C}_{3} ]$
Expanding along $R_1$ we get
$= 3 (x + y) {1 (2y^2 + y^2 )}$
$= 9y^2 (x + y) = RHS.$

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