What's the determinant of a 1x1 matrix?

Question

doc.brown · Answer

$$\left[ -4 \right]$$

anonymous · Answer

Ask Kai

kainui · Answer

Well I would just say that since: $$[-4]=-4*I_1$$ so it's really just the scalar -4 multiplied by the identity matrix, and the determinant of an identity matrix is just 1. so... -4.

You could think about it a little more in depth if you want too. For instance, when you break a nxn matrix determinant down into a bunch of (n-1)x(n-1) determinants.

kainui · Answer

Interesting question though, how did this come up? In your class or just curious? =)

anonymous · Answer

I like you in depth explanation haha

kainui · Answer

Well it's sort of vague and should let you think about it haha. I can always elaborate but I would rather have people ask questions about it based off what they think I mean. =P

doc.brown · Answer

I figured by cofactor expansion along the first row i would get +4 plus nothing else.

doc.brown · Answer

I was wondering if absolute values $|-4|$ were just mini determinants.

kainui · Answer

Nope, definitely not. When I said that I was more or less thinking about turning a 2x2 matrix into a couple determinants of 1x1 matrices like this by expanding across the top row.

$$\det \left[\begin{matrix}a & b\ c & d\end{matrix}\right]=a*\det[d]-b*\det[c]$$

doc.brown · Answer

but then det[d] is just d.  hmmm

doc.brown · Answer

Alright, you win.

kainui · Answer

Yup. =P