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MATH1850U/2050U:
Chapter 1 cont…
1
LINEAR SYSTEMS cont.
..
Inverses and Algebraic Properties of Matrices (1.4; pg. 38)
Recall:
Earlier today, we learned how to multiply matrices.
Caution:
There is no Commutative Law for matrix multiplication!
In other words, it is NOT
necessarily true that
AB
and
BA
will be equal.
Example:
Verify that
BA
AB
for the matrices below.
Luckily, though, most of the laws for real numbers also hold for matrices.
Theorem (Properties of Matrix Arithmetic):
Assuming the sizes of the matrices are such that
the indicated operations can be performed, the following rules of matrix arithmetic are valid.
a)
A
B
B
A
Commutative Law for Addition
b)
C
B
A
C
B
A
)
(
)
(
Associative Law for Addition
c)
C
AB
BC
A
)
(
)
(
Associative Law for Multiplication
d)
AC
AB
C
B
A
)
(
Left Distributive Law
e)
CA
BA
A
C
B
)
(
Right Distributive Law
f)
aC
aB
C
B
a
)
(
g)
bC
aC
C
b
a
)
(
h)
C
ab
bC
a
)
(
)
(
i)
)
(
)
(
)
(
aC
B
C
aB
BC
a
Definition:
A matrix whose entries are all zero is called a
zero matrix
.
Such a matrix is often
denoted by
0
or
n
m
0
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View Full Document MATH1850U/2050U:
Chapter 1 cont…
2
Theorem (Properties of Zero Matrices):
Assuming that the sizes of the matrices are such that
the indicated operations can be performed, the following rules of matrix arithmetic are valid.
a)
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This note was uploaded on 10/12/2011 for the course MATH 1020 taught by Professor Paulatu during the Spring '11 term at UOIT.
 Spring '11
 PaulaTu
 Algebra, Multiplication, Linear Systems, Matrices

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