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  • The value of the determinant is A. 9x2 (x + y)

The value of the determinant \left|\begin{array}{ccc} x & x+y & x+2 y \\ x+2 y & x & x+y \\ x+y & x+2 y & x \end{array}\right|  is
A. 9x^2 (x + y)
B. 9y^2 (x + y)
C. 3y^2 (x + y)
D. 7x^2 (x + y)

Answers (1)

B)

\\\begin{aligned} &\text { Matrix given: }\\ &\left|\begin{array}{ccc} x & x+y & x+2 y \\ x+2 y & x & x+y \\ x+y & x+2 y & x \end{array}\right|\\ &=\mathrm{x}\left|\begin{array}{cc} \mathrm{x} & \mathrm{x}+\mathrm{y} \\ \mathrm{x}+2 \mathrm{y} & \mathrm{x} \end{array}\right|-(\mathrm{x}+\mathrm{y})\left|\begin{array}{cc} \mathrm{x}+2 \mathrm{y} & \mathrm{x}+\mathrm{y} \\ \mathrm{x}+\mathrm{y} & \mathrm{x} \end{array}\right|+(\mathrm{x}+2 \mathrm{y})\left|\begin{array}{cc} \mathrm{x}+2 \mathrm{y} & \mathrm{x} \\ \mathrm{x}+\mathrm{y} & \mathrm{x}+2 \mathrm{y} \end{array}\right| \end{aligned}

\\\vspace{\baselineskip}= x [x\textsuperscript{2} -(x + y) (x + 2y)] -(x + y) [(x + 2y) (x) -(x + y)\textsuperscript{2}] + (x + 2y) [(x + 2y)\textsuperscript{2} -x (x + y)]\\ \\ \vspace{\baselineskip}= x [x\textsuperscript{2} -x\textsuperscript{2} -3xy -2y\textsuperscript{2}] -(x + y) [x\textsuperscript{2} + 2xy -x\textsuperscript{2} -2xy -y\textsuperscript{2}] + (x + 2y) [x\textsuperscript{2} + 4xy + 4y\textsuperscript{2} -x\textsuperscript{2} -xy]\\ \\ \vspace{\baselineskip}= x [-3xy -2y\textsuperscript{2}] -(x + y) [-y\textsuperscript{2}] + (x + 2y) [3xy + 4y\textsuperscript{2}]\\ \\ \vspace{\baselineskip}= -3x\textsuperscript{2}y -2xy\textsuperscript{2} + xy\textsuperscript{2} + y\textsuperscript{3} +3x\textsuperscript{2}y + 4xy\textsuperscript{2} + 6xy\textsuperscript{2} + 8y\textsuperscript{3}\\ \\ \vspace{\baselineskip}= 9y\textsuperscript{3} + 9xy\textsuperscript{2}\\ \\ \vspace{\baselineskip}= 9y\textsuperscript{2} (x + y)\\

 

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infoexpert22

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