MATRIX OPERATIONS
 Matrix Addition and Scalar Multiply:
 for
A, B and C
 A+B is defined by
(A+B)_{ij} = a_{ij} + b_{ij}.
 rA for scalar r is defined by
(rA)_{ij} = ra_{ij}.
 Properties
 1.
 A+B=B+A
 2.

(A+B)+C=A+(B+C)
 3.
 A+O=A, zero matrix, additive identity
 4.

r(A+B)=rA+rB, scalar r
 5.

(r+s)A=rA+sA, scalars r and s
 6.

r(sA)=(rs)A, scalars r and s
 Matrix Multiply:
 for an
A,
B and
C
 Definition
,
or
;
needs mnp scalar multiplications.
 Properties
 1.

A(BC) = (AB)C
 2.

A(B+C) = AB+AC
 3.

(B+C)A = BA+CA
 4.

r(AB) = (rA)B = A(rB) = (AB)r, scalar r
 5.

I_{m}A = AI_{n}, identity multiplication
 Commutativity:
 If AB = BA, then A and B are said to commute.
NOTE:
USUALLY!.
 Powers:

A^{k} = A^{k1}A; A^{0} = I, A^{1} = A, A^{2} = AA,
A^{3} = A^{2}A = AAA, etc.
Note: Compute
using
not
.
 Transpose:

(A^{T})_{ij} = (A)_{ji};
(AB)^{T} = B^{T}A^{T}.
 Scalar, Inner or Dot Product:

(sometimes denoted by
)
Note: if
is row i or A,
.
 Outer Product:


an ()
times a ()
is an
matrix,

 Outer product form for AB: if
is row i or B,
then
.
MATRIX INVERSES
 Inverse of A:
 For an
A
 Definition: if, for some
C,
AC = CA = I, then
A is invertible, and C is the inverse of A,
notated by A^{1}.

? If A is invertible, then
,
so
.
 Properties:
(A^{1})^{1} = A,
(AB)^{1} = B^{1}A^{1},
(A^{T})^{1}=(A^{1})^{T}.
 Finding A^{1}:
Alan C Genz
20000622