Rules for Matrix Algebra
MA 2071
1.� Matrix multiplication is, in general, not commutative.
That is, in general,� AB� ≠ �BA.��
What this means is that one should not change the order of matrices in
an expression, unlike ordinary algebra. (see examples from class 3/17)
2. Matrix
multiplication is linear.� This is a very
important concept! It amounts to , if c and k are
scalars,�
���������������������� A(cX� +� kY)� =� cAX
+ k AY�� �
����������� in other
words you can distribute
products and you can factor constants out.
����������� You have seen this concept in other
places:
����������������������� in
Calculus I, derivatives are linear��� d(c f(x)������
�+ k g(x))=� c df�� +�� kdg
����������������������������������������������������������������������������������� �dx�������������� ����������������������dx��������� dx�
����������������������� in
Calculus II, integrals
are linear
����������������������������������������������������������������������������������� 
����������� Also Fourier Transforms, important to Signal
Processing, are linear
����������������������������������������������������������� F( c(f(t)�
+� kg(t) )� = c F
(f(t))� +�
k F (g(t))
����������� and Laplace Transforms,
used in systems analysis, are linear
����������������������� ����������������������������������� L(
c(f(x)� +�
kg(x) )� = c L(f(x))�
+� k L(g(x))
3.� Matrix algebra is a binary operation. This means that at any time, no matter how many
matrices are involved
in an expression, you are only multiplying or
adding two at a time.� You have your
choice of which two, as long as
you do not reverse the order of
multiplication.� This is called the associative
property:
���������������� ������������������ ���(AB)C = A(BC) = ABC
4.� The identity matrix, In,� (1s on diagonal, 0s
elsewhere) serves as the multiplicative identity:
���������������� ������������������ AIn = A� ����and �������InA = A
���� (so this is an
example of a rare time when order does not matter).
5.� Some , but not all,
square matrices have multiplicative inverses. This means another matrix which
satisfies
���� AB
= BA = In
���� In such a case, B is
denoted by� A-1.� If a matrix has an inverse, it is
called nonsingular
in the text book.
���� Lots of matrices do not have inverses,
unlike ordinary numbers where only� 0 has no multiplicative inverse.
���� Those that do not are called singular
matrices.