Page 386 - 35Linear Algebra

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A B C D
A 1 1 1 0
B 1 1 1 1
C 1 1 1 0
D 0 1 0 1

Then take the entries of this table as a matrix
 
1 1 1 0
1 1 1 1
 
 
 1 1 1 0 
0 1 0 1
Notice that this table is symmetric across the diagonal, the same way a
multiplication table would be symmetric. This is because on facebook friend-
ship is symmetric in the sense that you can’t be friends with someone if they
aren’t friends with you too. This is an example of a symmetric matrix.
You could think about what you would have to do differently to draw a graph
for something like twitter where you don’t have to follow everyone who follows
you. The adjacency matrix might not be symmetric then.

Do Matrices Commute?

This video shows you a funny property of matrices. Some matrix properties
look just like those for numbers. For example numbers obey

a(bc) = (ab)c

and so do matrices:
A(BC) = (AB)C.
This says the order of bracketing does not matter and is called associativity.
Now we ask ourselves whether the basic property of numbers
ab = ba ,

holds for matrices
?
AB = BA .
For this, firstly note that we need to work with square matrices even for both
orderings to even make sense. Lets take a simple 2 × 2 example, let

1 a 1 b 1 0
A = , B = , C = .
0 1 0 1 a 1
In fact, computing AB and BA we get the same result

1 a + b
AB = BA = ,
0 1
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