I'm working out a few exercises for an exam, this is an interesting problem that should be simple (about 2 marks) but I can't seem to wrap my head around it. The question is:
Let $I$ be the $3\times 3$ Identity matrix. Show that: $$ \det(I_2(x))=x_2,\quad where \quad x=(x_1,x_2,x_3)\in\mathbb R^3 $$ (the $x$ is simply a vector in 3 dimensional space, I couldn't figure out how to do the correct notation on this site)
Now, I don't know what that subscript 2 of the Identity matrix means, but either it is the $2\times2$ Identity matrix which makes no sense or it means something else.
If it was a typing mistake where I got the question from and it was supposed to be a 3, then multiplying the $3\times3$ Identity matrix with $x$ and calculating the determinant gives me: $$\det(I_3(x))=x_1x_2x_3$$
Otherwise the secret might just be that the subscript 2 means calculating some sort of limited determinant?
Can anyone make sense of the question? Thank you in advance.