SOLUTION: when a question asks you to show something algebraically, DO NOT substitute numbers for the variables and give me an example. On the other hand, when a question asks you to show s

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Question 30537: when a question asks you to show something algebraically, DO NOT substitute numbers for the variables and give me an example. On the other hand, when a question asks you to show something numerically or to provide an example, you may substitute numbers for the variables. Show all work
Let A and B be two 2x2 matrices. Show by example that the determinant function is multiplicative, that is,
det(A)det(B) = det(AB)
(here you can pick numeric examples for A and B).
Bonus: show algebraically that the determinant function is multiplicative in the 2x2 case.

Answer by mbarugel(146) (Show Source):
You can put this solution on YOUR website!
I'll solve here the general case. We have two matrices:
,
Then we have that:
and
Now consider the multiplication A*B. The result of the multiplication is:

Now let's find the determinant of this matrix:

After applying the distributive property to each multiplication, we get:

But some of these terms cancel each other out. Specifically, we have and
We're left with

Now, let's check that this is the same as det(A)*det(B):

Applying distributive property:

Now compare the expressions we found for det(A*B) and for det(A)*det(B). Rearranging some of the factors in each term, they are exactly the same.
I hope this helps!
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