I'm taking an online linear algebra course and got stuck with a problem (it's not for credit)… Since I don't know anyone qualified in person, this is last resort. Pretty sure I've made some trivial error, but can't find it… If you can spot it, please let me know. Thank you in advance.
As we know, absolute value of determinant of a square matrix is equal to the volume of the parallelepiped with edges corresponding to the matrix's columns (or rows).
Consider matrix $A$ with columns $(1,1,0)^T,(0,1,1)^T,(1,0,1)^T$. Its determinant is 2. But it seems to me that the volume of the cube with edges corresponding to A's columns, is not 2… I mean, all three column vectors clearly have the same length, and it is $\sqrt2$, therefore volume of the cube with edges corresponding to these vectors must be $\sqrt2^3=2\sqrt2$.
What am I doing wrong?
Best Answer
The given vectors do not form a cube, so using the cube's volume formula is invalid. In particular, the vectors would need to be orthogonal (dot product zero for all pairs) for said formula to be applied, but here they are not. Indeed, the angle between each of the pairs here is 60°.