Applying rules to simplify a 3 x 3 determinant

[PcumP_Ravenclaw]

I have attached the properties/rules that will aid in simplifying the 3 x 3 determinant so that it can be calculated with minimum effort. I am trying to understand rule (2) and (4). I am not so clear about the idea of linearity. Can anyone please explain these two rules with regards to example 4.6.6 in the attachment?

Danke..

 

[DEvens]

For (2), what happens to the determinant if you multiply one row by a constant M.

Start with

A B C
D E F
G H J

Then multiply one row by M

A B C
MD ME MF
G H J

for example. What happens to the determinant? Linear means this multiplies the determinant by the same M.

As to (4), what it is saying is, if you add some multiple of row 2 to row 1, for example, it will not change the determinant. This is related to (3).