Section 3.1 – Matrix Operations
Pg 136, Definition – A
matrix
is a rectangular array of numbers called the
entries
, or
elements
of the matrix.
Pg 139, Definition – If A is an matrix and B is an
matrix, then the
product
C = AB is an
matrix. The
entry of the
product is computed as follows:
Pg 142, Theorem 3.1 – Let A be an
matrix,
a
standard unit vector, and
an
standard unit vector. Then
a.
is the ith row of A and
b.
is the jth column of A.
Pg 147, If A is a square matrix a r and s are nonnegative integers, then
1.
2.
Pg 149, Definition – The
transpose
of an
matrix A is the
matrix A
T
obtained by interchanging the rows and
columns of A. That is, the ith column of A
T
is the ith row of A for all i.
Pg 149, Definition – A square matrix A is
symmetric
if .
Pg 150, A square matrix A is symmetric iff
for all i and j.
Section 3.2 – Matrix Algebra
Pg 152, Theorem 3.2 –
Algebraic Properties of Matrix Addition and Scalar Multiplication
– Let A, B, and C be
matrices of the same size and let c and d be scalars. Then
a.
(Commutativity)
b.
(Associativity)
c.
d.
e.
(Distributivity)
f.
(Distributivity)
g.
h.
Pg 156, Theorem 3.3 –
Properties of Matrix Multiplication
– Let A, B, and C be matrices (whose sizes are such
that the indicated operations can be performed) and let k be a scalar. Then
a.
(Associativity)
b.
(Left distributivity)
c.
(Right distributivity)
d.
e.
(Multiplicative identity)
Pg 157, Theorem 3.4 –
Properties of the Transpose
– Let A and B be matrices (whose sizes are such that the
indicated operations can be performed) and let k be a scalar. Then
a.
b.
c.
d.
e.
Pg 159, Theorem 3.5
a.
If A is a square matrix, then
is a symmetric matrix.
b.
For any matrix A,
and
are symmetric matrices.
Section 3.3 – The Inverse of a Matrix
Pg 161, Definition – If A is an
matrix, an
inverse
of A is an
matrix A’ with the property that
where
is the
identity matrix. If such an A’ exists, then A is called
invertible
.
Pg 162, Theorem 3.6 – If A is an invertible matrix, then its inverse is unique.
Pg 163, Theorem 3.7 – If A is an invertible
matrix, then the system of linear equations given by
has the unique
solution
for any b in R
n
.
Pg 163, Theorem 3.8 – If , then is invertible if , in which case