In the following image,
How did they get from step (1) to (2)?
What does "Expanding across the first row" mean here?
This is called cofactor expansion. You can calculate the determinant by going down any column (the determinant of the transpose is the same, so you can also go down a row if you want.) Determinant Expansion by Minors -- from Wolfram MathWorld
The equation can be written as
$\displaystyle -\lambda (\lambda + 1) (\lambda - 1) + (\lambda + 1) + (\lambda + 1) = 0$
$\displaystyle -\lambda (\lambda + 1) (\lambda - 1) + 2(\lambda + 1) = 0$
Factorise this by taking out the common factor and simplifying.
Note: The final answer given in your image is wrong.
Ah thank you!
So it should be:
$\displaystyle (\lambda +1)(-\lambda^2+\lambda+2)=0$ ?
These lecture notes have many mistakes...
edit:
which is the same as:
$\displaystyle -(\lambda +1)(\lambda^2-\lambda-2)=0$
$\displaystyle \implies -(\lambda +1)(\lambda-2)(\lambda+1)=0$
$\displaystyle \implies -(\lambda +1)^2(\lambda-2)=0$
and so
$\displaystyle \lambda = -1, 2$ (has a repeated root).