## Search in the document preview

**Engineering Mathematics 3
Chapter 1
**

** MATRICES
**

Page **1** of **15** BTB28304

**1.0 MATRICES
**

A set of *m* x *n* numbers or functions, arranged in a rectangular array of *m* rows and *n*

column is called a matrix of order *m* x *n*, read as *m* by *n*.

To indicate the position of an element in a matrix, denote each element by a letter

followed by two suffixes.

A matrix of order *m* x* n* with double suffix notation is given by:

*mnmm
*

*n
*

*n
*

*aaa
*

*aaa
*

*aaa
*

...

....

...

...

21

22221

11211

The suffixes *i* and *j* in the element *i ja *indicate that the element *i ja *belongs to the *i
*th row

and *j *th column. Thus 23*a *belongs to second row and third column.

**1.1 TYPES OF MATRICES
**

a) ** Row matrix** – a matrix that has only one row.

Example:

741*A * Order of *A*: 1 x 3

b) ** Column matrix** – a matrix that has only one column.

Example:

2

5

7

*A * Order of *A*: 3 x 1

**Engineering Mathematics 3
Chapter 1
**

** MATRICES
**

Page **2** of **15** BTB28304

c) ** Square matrix** – a matrix which the numbers of row (

*m*) and column (

*n*) are equal.

The matrix is called a square matrix of order *n* x *n* or *n*.

Example:

6310

4869

1443

2135

*A * Order of *A*: 4 x 4 or 4

d) ** Diagonal matrix** – it is a square matrix of whose elements except those in the

leading diagonal are zero i.e. *i ja *= 0 when *ji * .

Example:

300

010

002

*A *

A diagonal matrix whose diagonal elements are all equal is called a *scalar matrix*.

Example:

200

020

002

*A *

A diagonal matrix whose diagonal elements are all equal to unity is called a *unit
*

*/identity* matrix denoted by *I*.

Example:

10

01
*I * and

100

010

001

*I *

e) ** Null /zero matrix** – a matrix whose elements are all zero.

Example:

000

000

000

*A *

**Engineering Mathematics 3
Chapter 1
**

** MATRICES
**

Page **3** of **15** BTB28304

f) ** Triangular matrix** – a square matrix whose elements either above or below the

leading diagonal are all zero.

Example:

**Upper** Triangular Matrix

33

2322

131211

00

0

*a
*

*aa
*

*aaa
*

*A *

**Lower** Triangular Matrix

333131

2221

11

0

00

*aaa
*

*aa
*

*a
*

*A *

g) ** MatrixTranspose**– a matrix obtained from any given matrix

*A*by changing rows

into columns or columns into rows and denoted by *TA *.

Example:

If

887

654

321

*A *, then

963

852

741
*TA *

Meanwhile, if

32 703

412

*B *then

23 74

01

32

*TB *

*transpose m x n matrix will change the order of matrix become n x m.

h) ** Symmetric matrix** – if a square matrix

*A*and its transpose

*TA*are identical such that

*AAT * .

Example:

705

034

541

*A * *TA *

**Engineering Mathematics 3
Chapter 1
**

** MATRICES
**

Page **4** of **15** BTB28304

i) ** Skew symmetric matrix** – if a square matrix

*A*such that

*AAT*

Example:

02

20
*A **AAT *

02

20

02

20

j) ** Singular matrix** – a matrix whose determinant

*A*is equal to zero.

If 0*A *, *A* is a non-singular matrix.

k) ** Adjoint matrix** – a matrix whose elements are the transpose of cofactor matrix.

l)* Equal matrix* – Two matrices

*A*and

*B*are said to be equal when they are of the same

order and the element in the corresponding positions are equal.

**1.2 MATRIX OPERATIONS
**

**1.2.1 Addition / Subtraction
**

Two matrices *A* and *B* can be added /subtracted only if **the order of A and B are
**

**the same order **(both have the same shape and size).

*Example 1:
*

Given the following matrices:

123

321
*A *

212

011
*B *

24

31
*C *

Solution:

131

312
*BA *

315

330
*BA *

The expression *CACBCA * ,, and *CB * are **undefined** since they have

different orders.

**Engineering Mathematics 3
Chapter 1
**

** MATRICES
**

Page **5** of **15** BTB28304

Note: In general, matrix addition is commutative that is

*ABBA *

*CBACBA * )()(

**1.2.2 Scalar Multiplication
**

For any matrix *A*, we can multiply *A* by a scalar, to form a new matrix of the same

order as *A*. This multiplication is performed by **multiplying every element of A by
**

**the scalar**.

*Example 2:
*

Let

42

03

21

*A *. Find *A*2 and *A
*3

1 .

Solution:

84

06

42

42

03

21

22*A *

3 4

3 2

01 3

2 3

1

42

03

21

3

1

3

1
*A *

**1.2.3 Matrix Multiplication
**

Two matrices *A* and *B* can be multiplied if and only if **matrix A has the same
**

**number of columns as B has in its rows** (i.e.

*A*is the order of

*p*x

*n*and

*B*is the

order of *n* x *q*). The resulting matrix *AB *will be of order *p* x *q*.

*Example 3:
*

If 31*C * and

520

312
*D *, can the product *CD* be formed?

**Commutative laws:
**- If changing the order of the

operands does not change the result.

- Swap the numbers over you still get the same result.

**Engineering Mathematics 3
Chapter 1
**

** MATRICES
**

Page **6** of **15** BTB28304

Solution:

*C* has size 21 . *D* has size 32 . Since the number of columns in *C* is the same as

the number of rows in *D*, we can form the product *CD*. The resulting matrix will have

size 31 as there is one row in *C* and three columns in *D*.

Suppose we wish to find *CD *when 31*C * and

520

312
*D *.

Then, )53()31())2(3()11()03()21( *CD *

1852 *CD *

*Example 4:
*

Given

062

421
*A * ,

7

1

1

*B *,

51

40

32

*C *,

31

21
,21 *ED *. determine

*AB * and *AC *. Solve for *BD* and *CE*

Solution:

7

1

1

062

421
*AB *

)7(0)1(6)1(2

)7(4)1(2)1(1

4

27

51

40

32

062

421
*AC *

)5(0)4(6)3(2)1(0)0(6)2(2

)5(4)4(2)3(1)1(4)0(2)2(1

304

276

*Note:

Matrix multiplication is not commutative, *BAAB *

Matrix multiplication is associative, )()( *BCACAB *

**Associative laws:
**- The order in which

the operations are performed does not matter as long as the sequence of the operands is not changed.

**Engineering Mathematics 3
Chapter 1
**

** MATRICES
**

Page **7** of **15** BTB28304

**1.3 MINOR AND COFACTORS
**

**1.2.2 Minor
**

Definition: Let *A* be an *n* x *n* matrix. The **minor***, ijM *, of the element *i ja *is the

determinant of the matrix obtained by deleting the* i*-th row vector and *j*-th column

vector of *A*.

Example:

If

333231

232221

131211

*aaa
*

*aaa
*

*aaa
*

*A *, then
3231

1211 23

*aa
*

*aa
M *

**1.2.3 Cofactor
**

Definition: Let *A* be an *n* x *n* matrix. The **cofactor***, i jA *, of the element *i ja *is defined by

*ij
ji
*

*ij MA
*

1 , where *ijM *is the minor of *i ja *.

The appropriate sign in the cofactor *i jA *, is easy to remember, since it alternates in

the following manner:

:::::

...

...

...

**1.4 DETERMINANT
**

Definition: the determinant of an *n* x *n* matrix *A*, denoted )det(*A * or *A *, is a scalar

associated with the matrix *A* that is defined as follows:

1 if ...

1 if )det(

1112121111

11

*nAaAaAa
*

*na
A
*

*nn
*

where *j
j
*

*j MA *1
1

1 det1

, *nj *,...,3,2,1

are the cofactor associated with the entries in the first row of *A*.

**Engineering Mathematics 3
Chapter 1
**

** MATRICES
**

Page **8** of **15** BTB28304

*Example 5:
*

If

652

420

132

*A *, find *A*det .

Solution:

Expand the determinant along the first row:

** ***Note: we can expand the determinant by any row or column in the same way and

obtain the same result in each case.

**1.4.1 Singular Matrix
**

Theorem: An *n* x *n* matrix is ** nonsingular** if and only if 0)det(

*A*.

If 0det *A *, then *A* is a ** singular** matrix.

*Example 6:
*

Determine whether the following matrices are nonsingular or singular.

753

242

311

*C * and

374

251

132

*D *

Solution:

*C* is a nonsingular matrix because 052)det( *C *.

*D* is a singular matrix because 0)det( *D *

**1.4.2 Use the Determinants to find Vector Products.
**

If **a** = *a*1**i** + *a*2**j** + *a*3**k** and **b** = *b*1**i** + *b*2**j **+ *b*3**k**, then

**a** x **b** =

321

321

*bbb
*

*aaa
*

*kji
*

44

42416

))]2(2()50[())]2(4()60[(3)]54()62[(2

52

20 )1(

62

40 3

65

42 2det

*A*

**Engineering Mathematics 3
Chapter 1
**

** MATRICES
**

Page **9** of **15** BTB28304

*Example 7:
*

If **a** = 3**i** + **j** - 2**k** and **b** = 4**i** + 5**k**, find** a** x **b**.

Solution:

**a** x **b** =

504

213

*kji
*

= 5**i **- 23**j** - 4**k
**

**1.5 PROPERTIES OF DETERMINANTS
**

Theorem: If *A* is an *n* x *n* upper or lower triangular matrix, then

*nnaaaaAA * ...det 332211

*Example 8:
*

Let

700

010

152

*A *. Find *A*det .

Solution:

Since *A* is an **upper triangular matrix**, then 14712det *A *.

**1.6 ELEMENTARY ROW OPERATIONS (ERO)
**

There are three operations for elementary row operations:

a) Interchange rows

b) Multiply a row by a nonzero constant

c) Add a multiple of one row to another row

Notations:

a) :RR *ji * interchange row* i* and row *j*

b) *ii k*RR : multiply row *i* by the scalar *k*

c) *jii k*RRR : add *k* times row *j* to row *i*

**Engineering Mathematics 3
Chapter 1
**

** MATRICES
**

Page **10** of **15** BTB28304

**1.6.1 Elementary Row Operations and Determinants
**

Let *A* be an *n* x *n* matrix.

**P1: **If *B* is a matrix obtained by interchanging two distinct rows of *A*, then

)det(det *BA *

**P2: **If *B* is the matrix obtained by dividing any row of *A* by a nonzero scalar *k*, then

)det(det *BkA *

**P3: **If *B* is the matrix obtained by adding a multiple of any row of *A* to a different row

of *A* then )det(det *BA *

*Example 9:
*

Let

243

421

312

*A *. Find *A*det by using the properties of determinants.

Solution:

243

312

421

:RR

243

421

312

21

by [**P1**]

243

530

421

:2RRR 122

by [**P3**]

10100

530

421

:3RRR 133

by [**P3**]

530

10100

421

:RR 32

by [**P1**]

**Engineering Mathematics 3
Chapter 1
**

** MATRICES
**

Page **11** of **15** BTB28304

530

110

421

10 :R 10

1 R 22

by [**P2**]

200

110

421

10 :R3RR 233

by [**P3**]

So, 2021110)det( *A *

**P4: **If *A* is an *n* x *n *matrix, then )det(det *AAT * .

**P5: **Let *A* and *B* be *n* x *n* matrices. Let *naaa *,...,, 21 denote the row vectors of *A*. If the

*i*-th row vector of *A *is the sum of two row vectors, for example *iii cba * , then

)det()det()det( *CBA *

where

*n
*

*i
*

*i
*

*i
*

*a
*

*a
*

*b
*

*a
*

*a
*

*B
*

:

:

1

1

1

and

*n
*

*i
*

*i
*

*i
*

*a
*

*a
*

*c
*

*a
*

*a
*

*C
*

:

:

1

1

1

.

The corresponding property is also true for columns.

**P6: **If *A* has a row(or column) of zeros, then 0det *A *.

**P7:** If two rows ( or columns) of *A* are the same, then 0det *A *.

**P8:** If *A* and *B* are matrices of the same size, then *BAAB *detdetdet

**P9:** Let *A* be an *n* x *n* matrix and *k* be a scalar, then *AkkA n *detdet

**P10: **Let *A* be an invertible matrix, then
)det(

1 det 1

*A
A * .

**MATRICES CHAPTER 1
**

Page **12** of **15** BTB28304

*Example 10:
*

Evaluate

1221

3335

2442

*x
*

*x
*

*x
*

.

Solution:

Apply **P5** to the first column:

122

333

244

121

335

242

*x
*

*x
*

*x
*

122

333

244

121

335

121

2

*x *

Since 31 RR and 21 CC

Thus, 0)0()0(2

1221

3335

2442

*x
*

*x
*

*x
*

*x
*

by **P7**.

**1.7 MATRIX INVERSE
**

If 0)det( *A *, *A* is a nonsingular matrix. Then, 1*A *exists.

**1.7.1 Steps to find the inverse of any matrix:**

1. Find *A*det

If 0)det( *A *, 1*A *does not exist. We stop all calculations.

If 0)det( *A *, we proceed to step 2

2. Find matrix of cofactor.

3. Find adjoint matrix, adj(*A*)

4. Find matrix inverse,

*A
A
*

det

11 adj(*A*)

**MATRICES CHAPTER 1
**

Page **13** of **15** BTB28304

*Example 11:
*

Find 1*A * if

134

421

203

*A *.

Solution:

1. Find *A*det :

32

134

421

203

Since 032det *A *, *A* is a nonsingular matrix so 1*A *exists.

2. Find the matrix of cofactor:

6144 956

51714

21

03

41

23

42

20

34

03

14

23

13

20

34

21

14

41

13

42

So, the matrix of cofactor is

6144

956

51714

3. Find adjoint matrix, adj(*A*):

adj(*A*)=

695

14517

4614

4. Find matrix inverse,

*A
A
*

det

11 adj(*A*)

32

11*A
*

695

14517

4614

**MATRICES CHAPTER 1
**

Page **14** of **15** BTB28304

**1.8 ROW TRANSFORMATION METHOD
**

Row transformation method is another method used to determine 1*A *.

*Example 12:
*

Use row transformation method to find 1*A * if

321

842

456

*A *.

Solution:

Start with I|*A *, where I is an identity matrix.

100:321

010:842

001:456

Use ERO:

001:456

010:842

100:321

:31 *RR *

001:456

210:1480

100:321

:2 12 *RR *

601:1470

210:1480

100:321

:6 13 *RR *

601:1470

41810:4710

100:321

: 8

1
2*R
*

417871:4700

41810:4710

100:321

:7 23 *RR *

7172174:100

41810:4710

100:321

: 7

4
3*R *

reduce the LHS to an identity matrix

**MATRICES CHAPTER 1
**

Page **15** of **15** BTB28304

7172174:100

411:010

100:321

: 4

7
32 *RR *

7172174:100

411:010

74423712:021

:3 31 *RR
*

7172174:100

411:010

7122172:001

:2 21 *RR *

**2
**

Therefore,

7172174

411

7122172
1*A *.

*Note: You should check this result by evaluating 1*AA *.

the required 1*A * on the RHS