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11

Matrices and Determinants

Case Study

11.1 Matrices

11.2 Determinants

11.3 Inverses of Square Matrices

Chapter Summary

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?

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 ?

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.

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.

11.1 Matrices

A. Introduction

Example 11.1T

Solution

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.

11.1 Matrices

B. Special Types of Matrices

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.

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).

11.1 Matrices

C. Operations of Matrices

Example 11.2T

Solution

? Y ? Z ? X ? Z ? X ? Y

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)

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).

11.1 Matrices

C. Operations of Matrices

Example 11.3T

Solution

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.

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.

11.1 Matrices

C. Operations of Matrices

Example 11.4T

Solution

11.1 Matrices

C. Operations of Matrices

Example 11.4T

Solution

(a)

11.1 Matrices

C. Operations of Matrices

Example 11.4T

Solution

YX is undefined.

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.

11.1 Matrices

C. Operations of Matrices

Example 11.5T

Solution

? AB ? 0

? BA ? 0

? c ? d ? 0

? b ? d ? 0

11.1 Matrices

C. Operations of Matrices

Remarks

The proofs are left for students.

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.

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

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

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.

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

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.

11.1 Matrices

C. Operations of Matrices

Example 11.8T

Solution

? ( At )2 ? pAt ? qI ? 0

11.2 Determinants

A. Introduction

11.2 Determinants

A. Introduction

Example 11.9T

Solution

11.2 Determinants

A. Introduction

To memorize the expansion of the determinant

? ? ? ? ? ?

Notes

This rule is only applicable for determinants of

order 3.

11.2 Determinants

A. Introduction

Example 11.10T

Solution

11.2 Determinants

A. Introduction

Example 11.11T

Solution

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.

11.2 Determinants

B. Properties of Determinants

11.2 Determinants

B. Properties of Determinants

When k ? 0, we have

When all the elements are also multiplied by k,

we have

11.2 Determinants

B. Properties of Determinants

In particular, we have

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

11.2 Determinants

B. Properties of Determinants

Finally, for the product of two square matrices,

we have

L.H.S.

R.H.S.

11.2 Determinants

B. Properties of Determinants

Example 11.12T

Solution

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.

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)

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)

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

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)

11.2 Determinants

C. Evaluation of Determinants of Order 3

Example 11.15T

Solution

11.2 Determinants

C. Evaluation of Determinants of Order 3

Example 11.16T

Solution

11.2 Determinants

C. Evaluation of Determinants of Order 3

Example 11.17T

Solution

11.2 Determinants

C. Evaluation of Determinants of Order 3

Example 11.18T

Solution

11.2 Determinants

C. Evaluation of Determinants of Order 3

Example 11.19T

Solution

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.

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.

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.

11.3 Inverses of Square Matrices

A. Introduction

By comparing the corresponding elements of the

matrices on both sides, we have

11.3 Inverses of Square Matrices

A. Introduction

11.3 Inverses of Square Matrices

A. Introduction

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.

11.3 Inverses of Square Matrices

A. Introduction

Example 11.20T

Solution

11.3 Inverses of Square Matrices

A. Introduction

Example 11.20T

Solution

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

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.

11.3 Inverses of Square Matrices

B. Properties of Inverses

Example 11.22T

Solution

11.3 Inverses of Square Matrices

B. Properties of Inverses

Example 11.22T

Solution

(b) (AB2)?1 ? (B2)?1A?1 ? (B?1)2A?1

11.3 Inverses of Square Matrices

B. Properties of Inverses

Example 11.22T

Solution

(AB)t?1 ? (AB)?1t ? (B?1A?1)t

(b)

11.3 Inverses of Square Matrices

B. Properties of Inverses

Example 11.23T

Solution

11.3 Inverses of Square Matrices

B. Properties of Inverses

Example 11.24T

Solution

11.3 Inverses of Square Matrices

B. Properties of Inverses

Example 11.24T

Solution

(b) Y ? (YX)X ?1

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

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

Chapter Summary

11.1 Matrices

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

Chapter Summary

11.2 Determinants

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.

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

Follow-up 11.2

11.1 Matrices

C. Operations of Matrices

Solution

? A ? B ? C ? C ? A ? B

Follow-up 11.3

11.1 Matrices

C. Operations of Matrices

Solution

Follow-up 11.4

11.1 Matrices

C. Operations of Matrices

Solution

Follow-up 11.4

11.1 Matrices

C. Operations of Matrices

Solution

(a)

Follow-up 11.4

11.1 Matrices

C. Operations of Matrices

Solution

QP is undefined.

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

Follow-up 11.6

11.1 Matrices

C. Operations of Matrices

Solution

A4 ? A2 A2

Follow-up 11.6

11.1 Matrices

C. Operations of Matrices

Solution

Follow-up 11.7

11.1 Matrices

C. Operations of Matrices

Solution

For n ? 1,

? The proposition is true for n ? 1.

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.

Follow-up 11.8

11.1 Matrices

C. Operations of Matrices

Solution

? ( At )2 ? pAt ? qI ? 0

Follow-up 11.9

11.2 Determinants

A. Introduction

Solution

Follow-up 11.10

11.2 Determinants

A. Introduction

Solution

Follow-up 11.11

11.2 Determinants

A. Introduction

Solution

Follow-up 11.12

11.2 Determinants

B. Properties of Determinants

Solution

Follow-up 11.13

11.2 Determinants

B. Properties of Determinants

Solution

Follow-up 11.13

11.2 Determinants

B. Properties of Determinants

Solution

? The given determinant is divisible by 25.

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)

Follow-up 11.15

11.2 Determinants

C. Evaluation of Determinants of Order 3

Solution

Follow-up 11.16

11.2 Determinants

C. Evaluation of Determinants of Order 3

Solution

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

Follow-up 11.17

11.2 Determinants

C. Evaluation of Determinants of Order 3

Solution

Follow-up 11.18

11.2 Determinants

C. Evaluation of Determinants of Order 3

Solution

Follow-up 11.19

11.2 Determinants

C. Evaluation of Determinants of Order 3

Solution

Follow-up 11.20

11.3 Inverses of Square Matrices

A. Introduction

Solution

Follow-up 11.20

11.3 Inverses of Square Matrices

A. Introduction

Solution

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

Follow-up 11.22

11.3 Inverses of Square Matrices

B. Properties of Inverses

Solution

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

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)

Follow-up 11.23

11.3 Inverses of Square Matrices

B. Properties of Inverses

Solution

Follow-up 11.24

11.3 Inverses of Square Matrices

B. Properties of Inverses

Solution

Follow-up 11.24

11.3 Inverses of Square Matrices

B. Properties of Inverses

Solution

(b) Q ? P ?1(PQ)

Follow-up 11.25

11.3 Inverses of Square Matrices

B. Properties of Inverses

Solution

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