09-240/Classnotes for Tuesday December 1

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

• These are "enough"!

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

• 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.