Download 1/6/2015 1 8.2 What You Should Learn Equality of Matrices Equality

1/6/2015
What You Should Learn
• Decide whether two matrices are equal.
8.2
OPERATIONS WITH MATRICES
• Add and subtract matrices and multiply matrices
by scalars.
• Multiply two matrices.
• Use matrix operations to model and solve
real-life problems.
Copyright © Cengage Learning. All rights reserved.
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Equality of Matrices
Equality of Matrices
Two matrices A = [aij] and B = [bij] are equal if they have
There is a rich mathematical theory of matrices, and its
applications are numerous. This section introduces some
fundamentals of matrix theory.
the same order (m ´ n) and aij = bij for 1 £ i £ m and
1 £ j £ n.
It is standard mathematical convention to represent
matrices in any of the following three ways.
In other words, two matrices are equal if their
corresponding entries are equal.
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Example 1 – Equality of Matrices
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Matrix Addition and Scalar Multiplication
Solve for a11, a12, a21, and a22 in the following matrix
equation.
In this section, three basic matrix operations will be
covered.
The first two are matrix addition and scalar multiplication.
With matrix addition, you can add two matrices (of the
same order) by adding their corresponding entries.
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Example 2 – Addition of Matrices
Example 2 – Addition of Matrices
cont’d
d. The sum of
and
is ___________ because A is of order 3 ´ 3 and B is of
order 3 ´ 2.
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Matrix Addition and Scalar Multiplication
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Matrix Addition and Scalar Multiplication
The symbol –A represents the negation of A, which is the
scalar product (–1)A.
In operations with matrices, numbers are usually referred to
as scalars. In this text, scalars will always be real
numbers.
Moreover, if A and B are of the same order, then A – B
represents the sum of A and (–1)B.
You can multiply a matrix A by a scalar c by multiplying
each entry in A by c.
That is,
A – B = A + (–1)B.
Subtraction of matrices
The order of operations for matrix expressions is similar to
that for real numbers.
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Example 3
é4
If A = ê 0
ê
êë- 3
- 1ù
4ú
ú
8 úû
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Matrix Addition and Scalar Multiplication
é 0 4ù
and B = ê- 1 3ú find:
ê
ú
êë 1 7úû
The properties of matrix addition and scalar multiplication
are similar to those of addition and multiplication of real
numbers.
a. -3A
b. 2B
c. -3A+2B
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Matrix Addition and Scalar Multiplication
Matrix Addition and Scalar Multiplication
Note that the Associative Property of Matrix Addition allows
you to write expressions such as A + B + C without
ambiguity because the same sum occurs no matter how
the matrices are grouped.
One important property of addition of real numbers is that
the number 0 is the additive identity. That is, c + 0 = c
for any real number c. For matrices, a similar property
holds.
This same reasoning applies to sums of four or more
matrices.
That is, if A is an m ´ n matrix and O is the m ´ n zero
matrix consisting entirely of zeros, then A + O = A.
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Matrix Addition and Scalar Multiplication
Matrix Addition and Scalar Multiplication
In other words, O is the additive identity for the set of all
m ´ n matrices.
The algebra of real numbers and the algebra of matrices
have many similarities.
For example, the following matrices are the additive
identities for the sets of all 2 ´ 3 and 2 ´ 2 matrices.
For example, compare the following solutions.
2 ´ 3 zero matrix
m ´ n Matrices
(Solve for X.)
Real Numbers
(Solve for x.)
and
2 ´ 2 zero matrix
x + a= b
X+A=B
x + a + (–a) = b + (–a)
X + A + (–A) = B + (–A)
x+0 =b–a
X+O=B–A
x=b–a
X=B–A
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Example 6
Matrix Multiplication
Another basic matrix operation is matrix multiplication.
Solve for X in the equation 2X-2A=B, where
A = é 0 3 ù and B = é- 4 - 8ù
ê2 0ú
ê
ú
ëê- 4 - 1ûú
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ê- 2 0 ú
ê
ú
ëê 14 - 6ûú
The definition of matrix multiplication indicates a
row-by-column multiplication, where the entry in the i th row
and jth column of the product AB is obtained by multiplying
the entries in the i th row of A by the corresponding entries
in the jth column of B and then adding the results.
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Example 7 – Finding the Product of Two Matrices
Matrix Multiplication
So for the product of two matrices to be defined, the
number of columns of the first matrix must equal the
number of rows of the second matrix.
Find the product AB using
and
The general pattern for matrix multiplication is as follows.
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Matrix Multiplication
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Example 9
Be sure you understand that for the product of two matrices
to be defined, the number of columns of the first matrix
must equal the number of rows of the second matrix.
Find the product and dimension of each of the following:
a. é 0 1 ù é 0 3 - 2ù
That is, the middle two indices must be the same. The
outside two indices give the order of the product, as shown
below.
A
´
B
=
AB
b. ê- 4
m´n
n´p
ê7 2ú ê 2 1
ë
ûë
é 2
ê
ëê 1
m´p
c.
4 úû
1ù
é3ù
0úú ê ú
-1
3úû ë û
é 0 2 4ù é 2 - 4 5ù
ê- 1 1 2 ú ê 0 9 3 ú
ë
ûë
û
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Matrix Multiplication
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Matrix Multiplication
Even if both AB and BA are defined, matrix multiplication is
not, in general, commutative. That is, for most matrices,
AB ¹ BA. This is one way in which the algebra of real
numbers and the algebra of matrices differ.
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Matrix Multiplication
Applications
If A is an n ´ n matrix, the identity matrix has the property
that AIn = A and In A = A. For example,
Matrix multiplication can be used to represent a system of
linear equations.
AI = A
Note how the system
can be written as the matrix equation AX = B, where A is
the coefficient matrix of the system, and X and B are
column matrices.
and
IA = A
A
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´
X
=
B
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Example 11 – Solving a System of Linear Equations
Consider the following system of linear equations.
x1 – 2x2 + x3 = – 4
x2 + 2x3 = 4
2x1 + 3x2 – 2x3 = 2
a. Write this system as a matrix equation, AX = B.
b. Use Gauss-Jordan elimination on the augmented matrix
to solve for the matrix X.
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