09-240/Classnotes for Tuesday December 1

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  A complete set of the December 1st lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

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~In the above gallery, there is a complete copy of notes for the lecture given on December 1st by Professor Natan (in PDF format).

--- Wiki Format ---

MAT240 – December 1st

Basic Properties of det: M_nxn→F: 0 det(I) = 1

1. $det(E'_{i,j\,\!}A) = -det(A) ; |E'_{i,j\,\!}|= -1. [Note: det(EA) = |E||A|]$

• Also, note that exchanging two rows flips the sign.

2. $det(E^2_{i,c\,\!}A) = det(A) ; |E^2_{i,j,c\,\!}| = 1$

• These are "enough"!

3. $det((E_{i,j,c\,\!}A) = det(A) ; |E^3_{i,j,c\,\!}| = 1$

• Adding a multiple of one row to another does not change the determinant.

The determinant of any matrix can be calculated using the properties above.

Theorem:

If $det' : M_{nxn\,\!}$→F satisfies properties 0-3 above, then $det' = det$

$det(A) = det'(A)$

Philosophical remark: Why not begin our inquiry with the properties above?

We must find an implied need for their use; thus, we must know whether a function $det$ exists first.