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In this section we’re going to take a quick look at some of
the properties of matrix arithmetic and of the transpose of a matrix. As mentioned in the previous section most of
the basic rules of real number arithmetic are still valid in matrix
arithmetic. However, there are a few
that are no longer valid in matrix arithmetic as we’ll be seeing.
We’ve already seen one of the real number properties that
doesn’t hold in matrix arithmetic. If a and b are two real numbers then we know by the commutative law for
multiplication of real numbers that 
(i.e.
(2)(3)=(3)(2)=6 ). However, if A and B are two matrices such that AB
is defined we saw an example in
the previous section in which BA was
not defined as well as an example
in which BA was defined and yet . In other words, we don’t have a commutative
law for matrix multiplication. Note that
doesn’t mean that we’ll never have for some matrices A and B, it is possible
for this to happen (as we’ll see in the next section) we just can’t guarantee
that this will happen if both AB and BA are defined.
Now, let’s take a quick look at the properties of real
number arithmetic that are valid in matrix arithmetic.
Properties
With real number arithmetic we didn’t need both 4. and 5.
since we’ve also got the commutative law for multiplication. However, since we don’t have the commutative
law for matrix multiplication we really do need both 4. and 5. Also, properties 6. 9. are simply distributive or associative laws
for dealing with scalar multiplication.
Now, let’s take a look at couple of other idea from real
number arithmetic and see if they have equivalent ideas in matrix arithmetic.
We’ll start with the following idea. From real number arithmetic we know that . Or, in other words, if we multiply a number
by 1 (one) doesn’t change the number.
The identity matrix will give
the same result in matrix multiplication.
If A is an matrix then we have,
Note that we really do need different identity matrices on
each side of A that will depend upon
the size of A.
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Example 1 Consider
the following matrix.
Then,
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Now, just like the identity matrix takes the place of the
number 1 (one) in matrix multiplication, the zero
matrix (denoted by 0 for a general matrix and 0 for a column/row matrix) will take the place of the number 0
(zero) in most of the matrix arithmetic.
Note that we said most of the matrix arithmetic. There are a couple of properties involving 0
in real numbers that are not necessarily valid in matrix arithmetic.
Let’s first start with the properties that are still valid.
Zero Matrix
Properties
Now, in real number arithmetic we know that if and then we must have (sometimes called the cancellation law). We also know that if then we have and/or (sometimes called the zero factor
property). Neither of these properties
of real number arithmetic are valid in general for matrix arithmetic.
We should be careful and not read too much into the results
of the previous example. The
cancellation law will not be valid in general for matrix multiplication. However, there are times when a variation of
the cancellation law will be valid as we’ll see in the next section.
Now, again, we need to be careful. There are times when we will have a variation
of the zero factor property, however there will be no zero factor property for
the multiplication of any two random matrices.
The next topic that we need to take a look at is that of
powers of matrices. At this point we’ll
just work with positive exponents. We’ll
need the next section before we can deal with negative exponents. Let’s start off with the following
definitions.
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Definition 1 If A is a square matrix then,
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We’ve also got several of the standard integer exponent
properties that we are used to working with.
Properties of Matrix
Exponents
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If A is a square
matrix and n and m are integers then,
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We can also talk about plugging matrices into polynomials
using the following definition. If we
have the polynomial,
and A is a square
matrix then,
where the identity matrix on the constant term has the same size as A.
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Example 4 Evaluate
each of the following for the given matrix.
(a)
(b)
(c) where
Solution
(a) There
really isn’t much to do with this problem.
We’ll leave it to you to verify the multiplication here.
(b) In this
case we may as well take advantage of the fact that we’ve got the result from
the first part already. Again, we’ll
leave it to you to verify the multiplication.
(c) In this
case we’ll need the result from the second part. Outside of that there really isn’t much to
do here.
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The last topic in this section that we need to take care of
is some quick properties of the transpose of a matrix.
Properties of the
Transpose
The first three of these properties should be fairly obvious
from the definition of the transpose.
The fourth is a little trickier to see, but isn’t that bad to verify.
Proof of #4 : We
know that the entry in the ith
row and jth column of AB is given by,
We also know that the entry in the ith row and jth
column of is found simply by interchanging the
subscripts i and j and so it is,
Now, let’s denote the entries of and as and respectively.
Again, based on the definition of the transpose we also know that,
and so from this we see that and . Finally, the entry in the ith row and jth column of is given by,
Now, plug in for and and we get that,
So, just what have we done here? We’ve managed to show that the entry in the ith row and jth column of is equal to the entry in the ith row and jth column of . Therefore, since each of the entries are equal
the matrices must also be equal.
Note that #4 can be naturally extended to more than two
matrices. For example,