Page 28 - A Course in Linear Algebra with Applications

12 Chapter One: Matrix Algebra

then the transpose of A is

A matrix which equals its transpose is called symmetric. On
the other hand, if A T equals —A, then A is said to be skew-
symmetric. For example, the matrices

are symmetric and skew-symmetric respectively. Clearly sym-
metric matrices and skew-symmetric matrices must be square.
We shall see in Chapter Nine that symmetric matrices can in
a real sense be reduced to diagonal matrices.
The laws of matrix algebra

We shall now list a number of properties which are sat-
isfied by the various matrix operations defined above. These
properties will allow us to manipulate matrices in a system-
atic manner. Most of them are familiar from arithmetic; note
however the absence of the commutative law for multiplica-
tion.
In the following theorem A, B, C are matrices and c, d are
scalars; it is understood that the numbers of rows and columns
of the matrices are such that the various matrix products and
sums mentioned make sense.
Theorem 1.2.1

(a) A + B = B + A, {commutative law of addition)]
(b) (A + B) + C = A + (B + C), (associative law of
addition);
(c) A + 0 = A;
(d) (AB)C = A(BC), ( associative law of multiplication)]
(e) AI = A = I A;