Fundamentals of Matrix Algebra, 2011a

Chapter 2
Matrix Arithmec
identy matrix is always a square matrix. We show a few identy matrices below.
⎡
⎤
⎡
[ ] 1 0
I 2 = , I
0 1 3 = ⎣ 1 0 0 ⎤ 1 0 0 0
0 1 0 ⎦ , I 4 = ⎢ 0 1 0 0
⎥
⎣ 0 0 1 0 ⎦
0 0 1
0 0 0 1
In our examples above, we have seen examples of things that do and do not work.
We should be careful about what examples prove, though. If someone were to claim
that AB = BA is always true, one would only need to show them one example where
they were false, and we would know the person was wrong. However, if someone
claims that A(B + C) = AB + AC is always true, we can’t prove this with just one
example. We need something more powerful; we need a true proof.
In this text, we forgo most proofs. The reader should know, though, that when
we state something in a theorem, there is a proof that backs up what we state. Our
jusficaon comes from something stronger than just examples.
Now we give the good news of what does work when dealing with matrix mulplicaon.
.
Ṫheorem 3
Properes of Matrix Mulplicaon
Let A, B and C be matrices with dimensions so that the following
operaons make sense, and let k be a scalar. The
following equalies hold:
.
1. A(BC) = (AB)C (Associave Property)
2. A(B + C) = AB + AB and
(B + C)A = BA + CA (Distribuve Property)
3. k(AB) = (kA)B = A(kB)
4. AI = IA = A
The above box contains some very good news, and probably some very surprising
news. Matrix mulplicaon probably seems to us like a very odd operaon, so we
probably wouldn’t have been surprised if we were told that A(BC) ≠ (AB)C. It is a
very nice thing that the Associave Property does hold.
As we near the end of this secon, we raise one more issue of notaon. We define
A 0 = I. If n is a posive integer, we define
A n =
}
A · A ·
{{
· · · · A
}
.
n mes
With numbers, we are used to a −n = 1 a n . Do negave exponents work with matrices,
too? The answer is yes, sort of. We’ll have to be careful, and we’ll cover the topic
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