Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

De determinant $\begin{vmatrix} x+y& x-y \\ x-y& x+y \\\end{vmatrix}$ is gelijk aan	A. 0
B. 2y²
C. − 2y²
D. 4xy
E. x² + y²

Evaluate $\begin{vmatrix} x+y& x-y \\ x-y& x+y \\\end{vmatrix}$	A. 0
B. 2y²
C. − 2y²
D. 4xy
E. x² + y²

1^ste manier :
$https://latex.codecogs.com/png.image?\dpi{110}\begin{vmatrix} x+y& x-y \\ x-y& x+y \\\end{vmatrix}=(x+y)^2-(x-y)^2=x^2+2xy+y^2-x^2+2xy-y^2=4xy\\$
2^de manier :
$https://latex.codecogs.com/png.image?\dpi{110}\\\begin{vmatrix} x+y& x-y \\ x-y& x+y \\\end{vmatrix}=(x+y)^2-(x-y)^2=(x+y+x-y)(x+y-x+y)\\=2x.2y=4xy$
3^de manier :
$https://latex.codecogs.com/png.image?\dpi{110}\\\begin{vmatrix} x+y& x-y \\ x-y& x+y \\\end{vmatrix}\overset{R_1\to R_1+R_2}{=}\begin{vmatrix}2x&2x\\x-y&x+y\\\end{vmatrix}=2\cdot \begin{vmatrix}x&x\\x-y&x+y\\\end{vmatrix}\\\overset{R_2\to R_2-R_1}{=}2\cdot \begin{vmatrix}x&x\\-y&+y\\\end{vmatrix}=2xy\begin{vmatrix}1&1\\-1&1\\\end{vmatrix}=2xy.2=4xy$
4^de manier :
$https://latex.codecogs.com/png.image?\dpi{110}\\\begin{vmatrix}x\!+\!y& x\!-\!y \\ x-y& x+y \\\end{vmatrix}\!=\!\begin{vmatrix}x&x\!-\!y\\x&x\!+\!y\\\end{vmatrix}\!+\!\begin{vmatrix}y&x\!-\!y\\-y&x\!+\!y\\\end{vmatrix}=\begin{vmatrix}x&x\\x&x\\\end{vmatrix}\!+\!\begin{vmatrix}x&-y\\x&y\\\end{vmatrix} \!+\!\begin{vmatrix}y&x\\-y&x\\\end{vmatrix} \!+\!\begin{vmatrix}y&-y\\-y&y\\\end{vmatrix} \\\\=0+2xy+2xy+0=4xy$