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In terms of the entries of A:
Main diagonal entries:___________________
Zero matrix :
Let A, B, and C be matrices of the same size, and let r and s
be scalars. Then
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 by defining
EXAMPLE: Compute AB where and
Note that Ab1 is a linear combination of the columns of A
Ab2 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
AB and BA?
BA would be
which is __________________.
|If A is m ·n and B is n ·p, then AB is m· p.|
Row-Column Rule for Computing AB (alternate method)
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
AB, if it is defined.
Solution: Since A is 2· 3 and B is 3 ·2, then AB is defined and
AB is _____× _____.
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)|
Properties above are analogous to properties of real numbers .
But NOT ALL real number properties correspond to matrix
1. It is not the case that AB always equal BA. (see Example 7,
2. Even if AB =AC, then B may not equal C. (see Exercise 10,
3. It is possible for AB =0 even if A≠ 0 and B ≠0. (see
Exercise 12, page 116)
If A is m ·n, the transpose of A is the n · m matrix,
AT, whose columns are formed from the corresponding rows of
EXAMPLE: Let Compute
Let A and B denote matrices whose sizes are appropriate for
the following sums and products.
a. (I.e., the transpose of AT is A
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,