# 09-240/Classnotes for Tuesday December 1

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

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MAT240 – December 1st

Basic Properties of $\det : \mathbb M_{n \times n} \rightarrow F$:

(Note that det(EA) = det(E)·det(A) and that det(A) may be written as |A|.)

0. $\,\! \det(I) = 1$

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

Exchanging two rows flips the sign.

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

These are "enough"!

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

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 ${\det}' : \mathbb M_{n \times n} \rightarrow F$ 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.