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Matrix Operations
Matrix Notation:
Two ways to denote m ·n matrix A:
In terms of the columns of A:
In terms of the entries of A:
Main diagonal entries:___________________
Zero matrix :
THEOREM 1
Let A, B, and C be matrices of the same size, and let r and s
be scalars. Then
Matrix Multiplication
Multiplying B and x transforms x into the vector Bx. In turn, if we
multiply A and Bx, we transform Bx into . So
is the
composition of two mappings .
Define the product AB so that
Suppose A is m ·n and B is n ·p where
and
Then
and
Therefore,
and by defining
we have
EXAMPLE: Compute AB where and
Solution :
Note that Ab_{1} is a linear combination of the columns of A
and
Ab_{2} is a linear combination of the columns of A.
Each column of AB is a linear combination of the
columns of A using weights from the corresponding columns of B. 
EXAMPLE: If A is 4 ·3 and B is 3 ·2, then what are the sizes
of
AB and BA?
Solution :
BA would be
which is __________________.
If A is m ·n and B is n ·p, then AB is m· p. 
RowColumn Rule for Computing AB (alternate method)
The definition
is good for theoretical work.
When A and B have small sizes, the following method is more
efficient when working by hand.
If AB is defined, let denote the entry in the ith row and jth
column of AB. Then
EXAMPLE Compute
AB, if it is defined.
Solution: Since A is 2· 3 and B is 3 ·2, then AB is defined and
AB is _____× _____.
THEOREM 2
Let A be m ·n and let B and C have sizes for which the
indicated sums and products are defined.
(associative law of multiplication )  
(left  distributive law )  
for any scalar r 
(right distributive law ) 
(identity for matrix multiplication) 
WARNINGS
Properties above are analogous to properties of real numbers .
But NOT ALL real number properties correspond to matrix
properties.
1. It is not the case that AB always equal BA. (see Example 7,
page 114)
2. Even if AB =AC, then B may not equal C. (see Exercise 10,
page 116)
3. It is possible for AB =0 even if A≠ 0 and B ≠0. (see
Exercise 12, page 116)
EXAMPLE:
If A is m ·n, the transpose of A is the n · m matrix,
denoted by
A^{T}, whose columns are formed from the corresponding rows of
A.
EXAMPLE:
EXAMPLE: Let Compute
and
Solution:
THEOREM 3
Let A and B denote matrices whose sizes are appropriate for
the following sums and products.
a. (I.e., the transpose of A^{T} is A
b.
c. For any scalar r,
d. (I.e. the transpose of a product of matrices
equals the product of their transposes in reverse order . )
EXAMPLE: Prove that =_________.
Solution: By Theorem 3d,
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