Question

prove that:det(x+y x x, 5x+4y 4x 2x, 10x+8y 8x 3x ) =x^3

Answer 1

$\text{Consider,}$

$\left|\begin{array}{ccc}x+y&x&x\\5x+4y&4x&2x\\10x+8y&8x&3x\end{array}\right|$

$=\left|\begin{array}{ccc}x+y&x&x\\5x+4y&4x&2x\\0&80&-x\end{array}\right|$ $R_3\implies\,R_3-2R_2$

$=\left|\begin{array}{ccc}x+y&x&0\\5x+4y&4x&2x\\0&80&-x\end{array}\right|$ $R_1\implies\,R_1+R_3$

$\text{Expanding along first row}$

$=(x+y)(-4x^2-0)-x(-5x^2-4xy)+0$

$=-4x^2(x+y)-x(-5x^2-4xy)$

$=-4x^3-4x^2y+5x^3+4x^2y$

$=-4x^3+5x^3$

$=x^3$

$\therefore\bf\left|\begin{array}{ccc}x+y&x&x\\5x+4y&4x&2x\\10x+8y&8x&3x\end{array}\right|=x^3$