Page 143 - 35Linear Algebra

7.3 Properties of Matrices 143

• Matrix operations on block matrices can be carried out by treating the
blocks as matrix entries. In the example above,

A B A B
2
M =
C D C D
2
A + BC AB + BD
=
CA + DC CB + D 2

Computing the individual blocks, we get:

 
30 37 44
2
A + BC =  66 81 96 
102 127 152
 
4
AB + BD =  
10
16

CA + DC = 4 10 16
CB + D 2 = (2)

Assembling these pieces into a block matrix gives:

 
30 37 44 4
66 81 96 10
 
 
102 127 152 16
 
4 10 16 2
2
This is exactly M .

7.3.3 The Algebra of Square Matrices

Not every pair of matrices can be multiplied. When multiplying two matrices,
the number of rows in the left matrix must equal the number of columns in
the right. For an r × k matrix M and an s × l matrix N, then we must
have k = s.
This is not a problem for square matrices of the same size, though.
Two n × n matrices can be multiplied in either order. For a single ma-
n
3
2
trix M ∈ M , we can form M = MM, M = MMM, and so on. It is useful
n
143