09-240/Classnotes for Tuesday December 1

--- Wiki Format ---

MAT240 – December 1st

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

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

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

2. [math]\displaystyle{ det(E^2_{i,c\,\!}A) = det(A) ; |E^2_{i,j,c\,\!}| = 1 }[/math]

• These are "enough"!

3. [math]\displaystyle{ det((E_{i,j,c\,\!}A) = det(A) ; |E^3_{i,j,c\,\!}| = 1 }[/math]

• 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 [math]\displaystyle{ det' : M_{nxn\,\!} }[/math]→F satisfies properties 0-3 above, then [math]\displaystyle{ det' = det }[/math]

[math]\displaystyle{ det(A) = det'(A) }[/math]

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 [math]\displaystyle{ det }[/math] exists first.