Module 2 Chapter 11 Matrices and Determinants

1
11
Matrices and Determinants
Case Study
11.1 Matrices
11.2 Determinants
11.3 Inverses of Square Matrices
Chapter Summary
2
Case Study
? Team X produce 500 pieces of product A, 200
pieces of product B and 350 pieces of product C
? Team Y produce 200 pieces of product A, 400
pieces of product B and 450 pieces of product C
Thats tedious!
Contents of ? product A 1.5 kg of copper, 0.2
kg of steel ? product B 0.6 kg of copper, 1.4
kg of steel ? product C 0.8 kg of copper, 1 kg
of steel
How to organize and calculate the total amount of
copper and steel needed by each team?
3
Case Study
Organization We can arrange the data in tabular
form
Product A Product B Product C
Team X 500 200 350
Team Y 200 400 450
Copper (in kg) Steel (in kg)
Product A 1.5 0.2
Product B 0.6 1.4
Product C 0.8 1
Calculation (1) Amount of copper needed by Team
X ?
(500 ? 1.5 ? 200 ? 0.6 ? 350 ? 0.8) kg
Copper Steel
Team X 1150 kg
Team Y

730 kg
900 kg 1050 kg
? 1150 kg
(2) Amount of steel needed by Team X ?
(3) Amount of copper needed by Team Y ?
(4) Amount of steel needed by Team Y ?
4
11.1 Matrices
A. Introduction
A rectangular array of numbers arranged in m rows
and n columns is called a m ? n matrix.
An m ? n matrix is represented in the form
or
A matrix with m rows and n columns is said to be
a matrix of order m ? n.
The number aij in the ith row and the jth column
of a matrix is called an element or entry.
5
11.1 Matrices
A. Introduction
For an a m ? n matrix, if m ? 1, it has only 1
row and is called a row matrix if n ? 1, it has
only 1 column and is called a column matrix.
( 5 4 3 ) is a row matrix of order 1 ? 3.
Two matrices are said to be equal if they satisfy
the following definition
Equality of Matrices Two matrices A ? (aij)m ? n
and B ? (bij)m ? n are equal if and only if they
have the same order and their corresponding
elements are equal, i.e., aij ? bij for all i
1, 2, 3, ... , m and j 1, 2, 3, ... , n.
6
11.1 Matrices
A. Introduction
Example 11.1T
Solution
7
11.1 Matrices
B. Special Types of Matrices
Zero Matrix A zero matrix, or a null matrix, is a
matrix that all its elements are zero.
Square Matrix A square matrix is a matrix with
the same numbers of rows and columns.
Notes
The order of a square matrix is denoted by its
number of rows n.
8
11.1 Matrices
B. Special Types of Matrices
9
11.1 Matrices
C. Operations of Matrices
Some rules on the operations of matrices
Addition of Matrices Suppose A ? (aij)m ? n and B
? (bij)m ? n are two matrices of order m ? n.
Then the sum of A and B is also an m ? n matrix C
? (cij)m ? n with cij ? aij ? bij, for all i ? 1,
2, 3, ... , m and j ? 1, 2, 3, ... , n.
10
11.1 Matrices
C. Operations of Matrices
Negative of Matrices Let A ? (aij)m ? n be an m ?
n matrix. The negative of A, denoted by ?A, is
the matrix whose elements are the negative of the
corresponding elements of A, i.e., ?A ? (?aij)m
? n, for all i ? 1, 2, 3, ... , m and j ? 1, 2,
3, ... , n.
Subtraction of Matrices Suppose A ? (aij)m ? n
and B ? (bij)m ? n are two matrices of order m ?
n. The difference of A and B is defined as A ? B
? A ? (?B).
11
11.1 Matrices
C. Operations of Matrices
Example 11.2T
Solution
? Y ? Z ? X ? Z ? X ? Y
12
11.1 Matrices
C. Operations of Matrices
Properties of Matrix Addition Let A ? (aij)m ? n,
B ? (bij)m ? n and C ? (cij)m ? n be m ? n
matrices and 0 be the m ? n zero matrix. Then we
have (a) A ? B ? B ? A (Commutative Law) (b) (A
? B) ? C ? A ? (B ? C) (Associative Law) (c) A ?
0 ? 0 ? A ? A (d) A ? (?A) ? (?A) ? A ? 0
Proofs of (a) and (b) By the definition of
addition of matrices,
A ? B ? (aij)m ? n ? (bij)m ? n
(A ? B) ? C ? (aij ? bij)m ? n ? (cij)m ? n
? (aij ? bij)m ? n
? (aij ? bij) ? cijm ? n
? (bij ? aij)m ? n
? aij ? (bij ? cij)m ? n
? (bij)m ? n ? (aij)m ? n
? (aij)m ? n ? (bij ? cij)m ? n
? B ? A
? A ? (B ? C)
13
11.1 Matrices
C. Operations of Matrices
Scalar Multiplication of Matrices The scalar
multiplication of an m ? n matrix A ? (aij)m ? n
and a real number k, which is denoted by kA, is
an m ? n matrix whose elements are the
corresponding elements of A multiplied by k,
i.e., kA ? (kaij)m ? n, for all i ? 1, 2, 3, ...
, m and j ? 1, 2, 3, ... , n.
Properties of Scalar Multiplication Let A and B
be two m ? n matrices and h, k be two real
numbers. We have (a) k(A ? B) ? kA ?
kB (Distributive Law) (b) (h ? k)A ? hA ?
kA (c) hkA ? h(kA) ? k(hA).
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11.1 Matrices
C. Operations of Matrices
Example 11.3T
Solution
15
11.1 Matrices
C. Operations of Matrices
Notes
When calculating the product AB, the matrix A
should be placed on the left while B is placed on
the right.
Multiplication of matrices is non-commutative,
i.e., for two matrices A and B, AB ? BA in
general.
16
11.1 Matrices
C. Operations of Matrices
? A is a 2 ? 3 matrix and B is a 3 ? 2 matrix.
? AB is a 2 ? 2 matrix.
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11.1 Matrices
C. Operations of Matrices
Example 11.4T
Solution
18
11.1 Matrices
C. Operations of Matrices
Example 11.4T
Solution
(a)
19
11.1 Matrices
C. Operations of Matrices
Example 11.4T
Solution
YX is undefined.
20
11.1 Matrices
C. Operations of Matrices
Even though A ? 0 and B ? 0, we still have AB ? 0
? AB ? 0 does not imply A ? 0 or B ? 0.
Consider AB ? AC
AB ? AC ? 0 A(B ? C) ? 0
? A ? 0 and B ? C.
? AB ? AC does not imply A ? 0 or B ? C ? 0.
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11.1 Matrices
C. Operations of Matrices
Example 11.5T
Solution
? AB ? 0
? BA ? 0
? c ? d ? 0
? b ? d ? 0
22
11.1 Matrices
C. Operations of Matrices
Remarks
The proofs are left for students.
23
11.1 Matrices
C. Operations of Matrices
For square matrices A and B of same order
1. (A ? B)2 ? (A ? B)(A ? B) ? AA ? AB ? BA ?
BB ? A2 ? AB ? BA ? B2
2. (A ? B)(A ? B) ? AA ? AB ? BA ? BB ? A2 ?
AB ? BA ? B2
In general, (A ? B)2 ? A2 ? 2AB ? B2 and (A ?
B)(A ? B) ? A2 ? B2.
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11.1 Matrices
C. Operations of Matrices
Example 11.6T
(a) Find the matrix X 2. (b) Hence, find the
matrix 3X 2 ? 2X ? 4I, where I is the 3 ? 3
identity matrix.
Solution
25
11.1 Matrices
C. Operations of Matrices
Example 11.6T
(a) Find the matrix X 2. (b) Hence, find the
matrix 3X 2 ? 2X ? 4I, where I is the 3 ? 3
identity matrix.
Solution
26
11.1 Matrices
C. Operations of Matrices
Example 11.7T
Solution
For n ? 1, obviously L.H.S. ? R.H.S. ? The
proposition is true for n ? 1.
When n ? k ? 1, L.H.S. ? X k ? 1
? R.H.S.
? The proposition is true for n ? k ? 1.
27
11.1 Matrices
C. Operations of Matrices
Transpose of Matrix Let A ? (aij)m ? n be an m ?
n matrix. The transpose of matrix of A, denoted
by At or AT, is an n ? m matrix At ? (cij)n ? m
such that cij ? aji for all i ? 1, 2, n and j ?
1, 2, , m.
The transpose of a matrix A is obtained by
interchanging the rows and the columns in A, for
examples
28
11.1 Matrices
C. Operations of Matrices
Properties of Transposes Let A and B be two m ? n
matrices, we have (a) (At)t ? A (b) (A ? B)t ?
At ? Bt (c) (kA)t ? kAt, where k is any
constant. Let A be an m ? n matrix and B be an n
? p matrix, we have (d) (AB)t ? BtAt.
Remarks
The proofs are left for students.
29
11.1 Matrices
C. Operations of Matrices
Example 11.8T
Solution
? ( At )2 ? pAt ? qI ? 0
30
11.2 Determinants
A. Introduction
31
11.2 Determinants
A. Introduction
Example 11.9T
Solution
32
11.2 Determinants
A. Introduction
To memorize the expansion of the determinant
? ? ? ? ? ?
Notes
This rule is only applicable for determinants of
order 3.
33
11.2 Determinants
A. Introduction
Example 11.10T
Solution
34
11.2 Determinants
A. Introduction
Example 11.11T
Solution
35
11.2 Determinants
B. Properties of Determinants
The following shows some of the properties of
determinants, which are true for determinants of
any order.
Remarks
These properties can be verified by expanding of
the determinants.
36
11.2 Determinants
B. Properties of Determinants
37
11.2 Determinants
B. Properties of Determinants
When k ? 0, we have
When all the elements are also multiplied by k,
we have
38
11.2 Determinants
B. Properties of Determinants
In particular, we have
39
11.2 Determinants
B. Properties of Determinants
Consider the result of addition of matrices, we
have
When p, q and r are proportional to the elements
of the other row, we have
40
11.2 Determinants
B. Properties of Determinants
Finally, for the product of two square matrices,
we have
L.H.S.
R.H.S.
41
11.2 Determinants
B. Properties of Determinants
Example 11.12T
Solution
42
11.2 Determinants
B. Properties of Determinants
Example 11.13T
Solution
Since all the elements in the determinant are
integers, its value in an integer. ? 3 is a
factor of the given determinant.
43
11.2 Determinants
C. Evaluation of Determinants of Order 3
The expansion of the determinant ? aei ? bfg ?
cdh ? ceg ? afh ? bdi
? a(ei ? fh) ? b( fg ? di) ? c(dh ? eg)
? a(ei ? fh) ? b(di ? fg) ? c(dh ? eg)
44
11.2 Determinants
C. Evaluation of Determinants of Order 3
The expansion of the determinant ? aei ? bfg ?
cdh ? ceg ? afh ? bdi
? b( fg ? di) ? e(ai ? cg) ? h(cd ? af )
? ?b(di ? fg) ? e(ai ? cg) ? h(af ? cd)
45
11.2 Determinants
C. Evaluation of Determinants of Order 3
Summarize the results as follows
Remarks
For each of the element, ? minor corresponding
determinant obtained ? cofactor product of the
minor and the sign of the term
46
11.2 Determinants
C. Evaluation of Determinants of Order 3
Example 11.14T
Solution
? ?1(?6 ? 48) ? 7(15 ? 72) ? 4(?30 ? 18)
? 4(?30 ? 18) ? 8(?6 ? 63) ? 3(?2 ? 35)
47
11.2 Determinants
C. Evaluation of Determinants of Order 3
Example 11.15T
Solution
48
11.2 Determinants
C. Evaluation of Determinants of Order 3
Example 11.16T
Solution
49
11.2 Determinants
C. Evaluation of Determinants of Order 3
Example 11.17T
Solution
50
11.2 Determinants
C. Evaluation of Determinants of Order 3
Example 11.18T
Solution
51
11.2 Determinants
C. Evaluation of Determinants of Order 3
Example 11.19T
Solution
52
11.3 Inverses of Square Matrices
A. Introduction
For matrices, matrix division is not defined.
We can try to find a matrix B such that BA ? AB ?
I.
Inverse of a Matrix If square matrices A and B of
order n satisfy the relationship AB BA I,
where I is the identity matrix of order n, then
the matrix B is called the inverse of A and
denoted by A?1, i.e., AA?1 ? A?1A ? I.
53
11.3 Inverses of Square Matrices
A. Introduction
For example, consider
? B is the inverse of A and A is the inverse of B.
In particular, the inverse of an identity matrix
is the identity matrix itself.
54
11.3 Inverses of Square Matrices
A. Introduction
Actually, not all square matrices have their
corresponding inverses.
Singular and Non-singular Matrices A square
matrix A is said to be non-singular or invertible
if and only if its inverse exists. Otherwise, it
is said to be singular or non-invertible.
If the inverse of a square matrix exists, then we
have
Uniqueness of Inverse The inverse of a
non-singular square matrix is unique.
Proof (using contraction) Suppose B and C are
two distinct inverse matrices of A, i.e., AB ? BA
? I and AC ? CA ? I.
Then B ? BI
? BAC
? IC
? C, which contradicts to B ? C.
55
11.3 Inverses of Square Matrices
A. Introduction
By comparing the corresponding elements of the
matrices on both sides, we have
56
11.3 Inverses of Square Matrices
A. Introduction
57
11.3 Inverses of Square Matrices
A. Introduction
58
11.3 Inverses of Square Matrices
A. Introduction
Proof if If A is non-singular, then there
exists a matrix B such that AB ? BA ? I.
? A is non-singular.
59
11.3 Inverses of Square Matrices
A. Introduction
Example 11.20T
Solution
60
11.3 Inverses of Square Matrices
A. Introduction
Example 11.20T
Solution
61
11.3 Inverses of Square Matrices
A. Introduction
Example 11.21T
Let P be a square matrix such that 2I ? P ? P2 ?
0. Prove that P is non-singular and find P?1 in
terms of P and I.
Solution
2I ? P ? P2 ? 0 P ? P2 ? 2I
P(I ? P) ? 2I
62
11.3 Inverses of Square Matrices
B. Properties of Inverses
Proof of (f) ? (AB)(B?1A?1) ? A(BB?1)A?1
? AIA?1
? AA?1 ? I and
(B?1A?1)(AB) ? B?1(A?1A)B ? B?1IB ? B?1B ? I
? By definition, (AB)?1 ? B?1A?1.
63
11.3 Inverses of Square Matrices
B. Properties of Inverses
Example 11.22T
Solution
64
11.3 Inverses of Square Matrices
B. Properties of Inverses
Example 11.22T
Solution
(b) (AB2)?1 ? (B2)?1A?1 ? (B?1)2A?1
65
11.3 Inverses of Square Matrices
B. Properties of Inverses
Example 11.22T
Solution
(AB)t?1 ? (AB)?1t ? (B?1A?1)t
(b)
66
11.3 Inverses of Square Matrices
B. Properties of Inverses
Example 11.23T
Solution
67
11.3 Inverses of Square Matrices
B. Properties of Inverses
Example 11.24T
Solution
68
11.3 Inverses of Square Matrices
B. Properties of Inverses
Example 11.24T
Solution
(b) Y ? (YX)X ?1
69
11.3 Inverses of Square Matrices
B. Properties of Inverses
Example 11.25T
(a) Find the matrix Y ?1XY. (b) Hence find X 1000.
Solution
70
11.3 Inverses of Square Matrices
B. Properties of Inverses
Example 11.25T
(a) Find the matrix Y ?1XY. (b) Hence find X 1000.
Solution
(b) Consider (Y ?1XY)1000 ? (Y ?1XY)(Y ?1XY)(Y
?1) ( Y)(Y ?1XY)
? Y ?1X(I) X(I) (I) XY
? Y ?1X 1000Y
? Y(Y ?1XY)1000Y ?1 ? X 1000
71
Chapter Summary
11.1 Matrices
72
Chapter Summary
11.1 Matrices
2. Operations of Matrices Let A ? (aij)m ? n and
B ? (bij)m ? n be two matrices and k be a real
number.
(a) Addition A ? B ? (aij ? bij)m ? n, for all
i ? 1, 2, ... , m and j ? 1, 2, ... , n
(b) Subtraction A ? B ? (aij ? (?1)bij)m ? n,
for all i ? 1, 2, ... , m and j ? 1, 2, ... , n
(c) Scalar Multiplication kA ? (kaij)m ? n
(d) Transpose At ? (cij)n ? m where cij ? aji,
for all i ? 1, 2, ... , n and j ? 1, 2, ... , m
73
Chapter Summary
11.2 Determinants
74
Chapter Summary
11.3 Inverses of Square Matrices
1. Definition For a square matrix A, if there
exists a matrix B such that AB ? BA ?
I, then B is called the inverse of A and is
denoted by A?1.
75
Follow-up 11.1
11.1 Matrices
A. Introduction
If ( 3 5 7 ) ? ( i j k ), find the
values of i, j and k.
Solution
76
Follow-up 11.2
11.1 Matrices
C. Operations of Matrices
Solution
? A ? B ? C ? C ? A ? B
77
Follow-up 11.3
11.1 Matrices
C. Operations of Matrices
Solution
78
Follow-up 11.4
11.1 Matrices
C. Operations of Matrices
Solution
79
Follow-up 11.4
11.1 Matrices
C. Operations of Matrices
Solution
(a)
80
Follow-up 11.4
11.1 Matrices
C. Operations of Matrices
Solution
QP is undefined.
81
Follow-up 11.5
11.1 Matrices
C. Operations of Matrices
Solution
? AB ? 0
? BA ? 0
? a ? b ? 0
? a ? b ? 0 and c ? d ? 0
b ? ?a and d ? ?c
82
Follow-up 11.6
11.1 Matrices
C. Operations of Matrices
Solution
A4 ? A2 A2
83
Follow-up 11.6
11.1 Matrices
C. Operations of Matrices
Solution
84
Follow-up 11.7
11.1 Matrices
C. Operations of Matrices
Solution
For n ? 1,
? The proposition is true for n ? 1.
85
Follow-up 11.7
11.1 Matrices
C. Operations of Matrices
Solution
When n ? k ? 1, L.H.S. ? M k ? 1
? The proposition is true for n ? k ? 1.
? By the principle of mathematical induction, the
proposition is true for all positive integers n.
86
Follow-up 11.8
11.1 Matrices
C. Operations of Matrices
Solution
? ( At )2 ? pAt ? qI ? 0
87
Follow-up 11.9
11.2 Determinants
A. Introduction
Solution
88
Follow-up 11.10
11.2 Determinants
A. Introduction
Solution
89
Follow-up 11.11
11.2 Determinants
A. Introduction
Solution
90
Follow-up 11.12
11.2 Determinants
B. Properties of Determinants
Solution
91
Follow-up 11.13
11.2 Determinants
B. Properties of Determinants
Solution
92
Follow-up 11.13
11.2 Determinants
B. Properties of Determinants
Solution
? The given determinant is divisible by 25.
93
Follow-up 11.14
11.2 Determinants
C. Evaluation of Determinants of Order 3
Solution
? ?2(63 ? 12) ? 8(9 ? 24) ? 5(3 ? 42)
? ?7(?30 ? 18) ? 3(5 ? 8) ? 8(9 ? 24)
94
Follow-up 11.15
11.2 Determinants
C. Evaluation of Determinants of Order 3
Solution
95
Follow-up 11.16
11.2 Determinants
C. Evaluation of Determinants of Order 3
Solution
96
Follow-up 11.17
11.2 Determinants
C. Evaluation of Determinants of Order 3
c2 ? (a ? b)2 ? (c ? a ? b)(c ? a ? b)
(b ? c)2 ? a2 ? (b ? c ? a)(b ? c ? a)
Solution
97
Follow-up 11.17
11.2 Determinants
C. Evaluation of Determinants of Order 3
Solution
98
Follow-up 11.18
11.2 Determinants
C. Evaluation of Determinants of Order 3
Solution
99
Follow-up 11.19
11.2 Determinants
C. Evaluation of Determinants of Order 3
Solution
100
Follow-up 11.20
11.3 Inverses of Square Matrices
A. Introduction
Solution
101
Follow-up 11.20
11.3 Inverses of Square Matrices
A. Introduction
Solution
102
Follow-up 11.21
11.3 Inverses of Square Matrices
A. Introduction
Let X be a square matrix such that 2X 2 ? 4X ? 5I
? 0. Prove that X is non-singular and find X ?1
in terms of X and I.
Solution
2X 2 ? 4X ? 5I ? 0 2X 2 ? 4X ? 5I
2X (X ? 2I ) ? 5I
103
Follow-up 11.22
11.3 Inverses of Square Matrices
B. Properties of Inverses
Solution
104
Follow-up 11.22
11.3 Inverses of Square Matrices
B. Properties of Inverses
Solution
(b) (M 2N)?1 ? N ?1(M 2)?1 ? N ?1(M ?1)2
105
Follow-up 11.22
11.3 Inverses of Square Matrices
B. Properties of Inverses
Solution
(NM t)?1 ? (M t)?1N ?1 ? (M ?1)tN ?1
(b)
106
Follow-up 11.23
11.3 Inverses of Square Matrices
B. Properties of Inverses
Solution
107
Follow-up 11.24
11.3 Inverses of Square Matrices
B. Properties of Inverses
Solution
108
Follow-up 11.24
11.3 Inverses of Square Matrices
B. Properties of Inverses
Solution
(b) Q ? P ?1(PQ)
109
Follow-up 11.25
11.3 Inverses of Square Matrices
B. Properties of Inverses
Solution
110
Follow-up 11.25
11.3 Inverses of Square Matrices
B. Properties of Inverses
Solution
(b) Consider (P ?1QP)800 ? (P ?1QP)(P ?1QP)(P
?1) ( P)(P ?1QP)
? P ?1Q(I) Q(I) (I) QP
? P ?1Q800P
? P(P ?1QP)800P ?1 ? Q 800