Fundamentals of Matrix Algebra, 2011a

Chapter 2
Matrix Arithmec
BB = BC, yet B ≠ C. In general, just because AX = BX, we cannot conclude that A = B.
Matrix mulplicaon is turning out to be a very strange operaon. We are very
used to mulplying numbers, and we know a bunch of properes that hold when using
this type of mulplicaon. When mulplying matrices, though, we probably find ourselves
asking two quesons, “What does work?” and “What doesn’t work?” We’ll answer
these quesons; first we’ll do an example that demonstrates some of the things
that do work.
. Example 30 .Let
A =
[ ] [ ]
1 2 1 1
, B =
3 4 1 −1
and C =
[ ] 2 1
.
1 2
Find the following:
1. A(B + C)
2. AB + AC
3. A(BC)
4. (AB)C
S We’ll compute each of these without showing all the intermediate
steps. Keep in mind order of operaons: things that appear inside of parentheses
are computed first.
1.
[ ] ([ 1 2 1 1
A(B + C) =
3 4 1 −1
[ ] [ ]
1 2 3 2
=
3 4 2 1
[ ] 7 4
=
17 10
]
+
[ ]) 2 1
1 2
2.
[ ] [ 1 2 1 1
AB + AC =
3 4 1 −1
[ ] 3 −1
= +
7 −1
[ ] 7 4
=
17 10
]
+
[ 4 5
10 11
[ 1 2
3
]
4
] [ ] 2 1
1 2
60