Page 141 - 35Linear Algebra

7.3 Properties of Matrices 141

Sometimes matrices do not share the properties of regular numbers. In
particular, for generic n × n square matrices M and N,

MN 6= NM .

Do Matrices Commute?

Example 88 (Matrix multiplication does not commute.)

1 1 1 0 2 1
=
0 1 1 1 1 1
while, on the other hand,

1 0 1 1 1 1
= .
1 1 0 1 1 2
n
n
Since n × n matrices are linear transformations R → R , we can see that
the order of successive linear transformations matters.
Here is an example of matrices acting on objects in three dimensions that
also shows matrices not commuting.

Example 89 In Review Problem 3, you learned that the matrix

cos θ sin θ
M = ,
− sin θ cos θ
rotates vectors in the plane by an angle θ. We can generalize this, using block matrices,
to three dimensions. In fact the following matrices built from a 2 × 2 rotation matrix,
a 1 × 1 identity matrix and zeroes everywhere else

   
cos θ sin θ 0 1 0 0
M =  − sin θ cos θ 0  and N =  0 cos θ sin θ  ,
0 0 1 0 − sin θ cos θ
perform rotations by an angle θ in the xy and yz planes, respectively. Because, they
rotate single vectors, you can also use them to rotate objects built from a collection of
vectors like pretty colored blocks! Here is a picture of M and then N acting on such
◦
a block, compared with the case of N followed by M. The special case of θ = 90 is
shown.

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