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. [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: