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 :
Let A, B, and C be matrices of the same size, and let r and s be scalars.
Multiplying B and x transforms x into the vector Bx. In turn, if we multiply
A and Bx, we transform
Bx into A(Bx) . So A(Bx) is the composition of two mappings.
Define the product AB so that A(Bx)=(AB)x.
Suppose A is m*n and B is n*p where
and by defining
we have A(Bx)=(AB)x.
EXAMPLE: Compute AB where
Note that Ab1 is a linear combination of the columns of A and 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 of AB and
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 (AB)ij denote the entry in the ith row and jth column of
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
Properties above are ana logous to properties of real numbers . But NOT ALL
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)
Powers of A
If A is m*n, the transpose of A is the n*m matrix, denoted by AT, whose
columns are formed
from the corresponding rows of A.
Let Compute AB,
Let A and B denote matrices whose sizes are appropriate
for the fol lowing sums and
a. (I.e., the transpose
of AT is A
c. For any scalar r,
(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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