Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

De determinant $https://latex.codecogs.com/png.image?\dpi{110}\begin{vmatrix}a&a&b+1\\b&1&a\\2&b+1&1\\\end{vmatrix}$ is te ontbinden in precies drie factoren. Eén ervan is (a + b + 2). De twee andere zijn	A. (b − 1)(1 − a + b)
	B. (b − 1)(b −1− a)
	C. (b + 1)(b − a + 1)
	D. (b + 1)(b − a − 1)
	E. (a − b)(1 − b)

[ 5-6118 - op net sinds 6.2.13-(E)-15.11.2024 ]

Translation in E N G L I S H

Factoring the determinant $https://latex.codecogs.com/png.image?\dpi{110}\begin{vmatrix}a&a&b+1\\b&1&a\\2&b+1&1\\\end{vmatrix}$ one of the factors is (a + b + 2). The two other factors are	A. (b−1)(1− a+b)
	B. (b−1)(b−1− a)
	C. (b+1)(b− a+1)
	D. (b+1)(b− a−1)
	E. (a− b)(1− b)

Oplossing - Solution

$\begin{aligned} & \left|\begin{matrix}a&a&b+1\\b&1&a\\2&b+1&1\\\end{matrix}\right| =\left|\begin{matrix}a+b+2&a+b+2&a+b+2\\b&1&a\\2&b+1&1\\\end{matrix}\right| =(a+b+2)\left|\begin{matrix}1&1&1\\b&1&a\\2&b+1&1\\\end{matrix}\right|\\ & =(a+b+2)\left|\begin{matrix}1&0&0\\b&1-b&a-b\\2&b-1&-1\\\end{matrix}\right| =(a+b+2)\left|\begin{matrix}1-b&a-b\\b-1&-1\end{matrix}\right|\\ & =(a+b+2)(b-1)\left|\begin{matrix}-1&a-b\\1&-1\end{matrix}\right| =(a+b+2)(b-1)(1-a+b) \end{aligned}$