Page 37 - A Course in Linear Algebra with Applications
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1.2: Operations with Matrices 21
A=(A 1\A 2\ ... \A n).
Because of this it is important to observe the following fact.
Theorem 1.2.4
Partitioned matrices can be added and multiplied according to
the usual rules of matrix algebra.
Thus to add two partitioned matrices, we add correspond-
ing entries, although these are now matrices rather than
scalars. To multiply two partitioned matrices use the row-
times-column rule. Notice however that the partitions of the
matrices must be compatible if these operations are to make
sense.
Example 1.2.11
Let A = (0^)4,4 be partitioned into four 2 x 2 matrices
A = A n A 12
A 2\ A22
where
An = ( a n G l 2 ) , A 12=l ° 1 3 a i 4
«21 &22 ) \ 023 «24
= ( a 3 1 a 3 2 =
•4 21 ) , A 2 2
a
V 4i a 42 J
Let B = (fry)4,4 be similarly partitioned into submatrices Bn,
B\2, B21, B22
Bn B\2
B
B 2\ B22
T h e n
+ +
A + B A n B n A 12 B 12
A21 + B21 A22 + B22
