Jacobian matrix, algebra help

Question

I am just a little confused with some algebra for a jacobian matrix,

I have $ \begin {pmatrix} 1 - \frac{2}{1+\alpha} -\frac{\alpha}{1+\alpha} &- \frac{\alpha}{1+\alpha} \\ \beta & -2\beta \end{pmatrix}$

for the top part of the matrix, why is it wrong to simplyfy like this

$1(1+\alpha)- \frac{2}{1+\alpha}(1+\alpha) -\frac{\alpha}{1+\alpha}(1+\alpha) $

which simplifies to $-1$, however in my teacher shows it simplifies to $ -\frac{1}{1+\alpha}$

how and wh y ?

Answer 1

Note that

$$1 - \frac{2}{1+\alpha} -\frac{\alpha}{1+\alpha}=\frac{1+\alpha-2-\alpha}{1+\alpha}=\frac{-1}{1+\alpha} $$

The first method could be used if we are calculating the jacobian determinant but we need to multiply also the second entry of the row by $(1+\alpha)$ and to divide, at the end, the final jacobian determinant by $(1+\alpha)$.

Answer 2

$$ 1 - \frac{2}{1+\alpha} -\frac{\alpha}{1+\alpha}= \frac{1+\alpha}{1+\alpha} - \frac{2}{1+\alpha} -\frac{\alpha}{1+\alpha}= \frac{1+\alpha-2-\alpha}{1+\alpha} $$