Engineering Mathematics 4 Episode 14 pdf

iii The value of a 3 by 3 determinant is the sum of the products of the elements and their cofactors of any row or any column of the corresponding 3 by 3 matrix.. Determine the value ofN

60.5 The inverse or reciprocal of a 2

by 2 matrix

The inverse of matrix A is A1such that AðA1DI,

the unit matrix

There is, however, a quicker method of obtaining

the inverse of a 2 by 2 matrix.

For any matrix

(i) interchanging the positions of p and s,

(ii) changing the signs of q and r, and

(iii) multiplying this new matrix by the reciprocal

inter-of q and r and multiplying by the reciprocal inter-of thedeterminantp q

1 13

−7 26

3 26





Now try the following exercise

Exercise 203 Further problems on the

117417

317

23

1

35

42

37

3 Determine the inverse of

(i) The minor of an element of a 3 by 3 matrix is

the value of the 2 by 2 determinant obtained

by covering up the row and column containing

of element 4 is obtained by covering the row

(4 5 6) and the column



147



, leaving the 2 bydeterminant2 3



, i.e the minor of element

(ii) The sign of a minor depends on its position

within the matrix, the sign pattern being

Thus the signed-minor of

element 4 in the matrix

The signed-minor of an element is called the

cofactor of the element.

(iii) The value of a 3 by 3 determinant is the

sum of the products of the elements and

their cofactors of any row or any column

of the corresponding 3 by 3 matrix.

There are thus six different ways of evaluating a

3 ð 3 determinant — and all should give the same

Supposing a second row expansion is selected.The minor of 2 is the value of the determinantremaining when the row and column containingthe 2 (i.e the second row and the first column),

is covered up Thus the cofactor of element 2 is



34 12i.e 11 The sign of element 2 is minus,

(see (ii) above), hence the cofactor of element 2, (thesigned-minor) is C11 Similarly the minor of ele-ment 7 is3 4



i.e 13, and its cofactor is C13.Hence the value of the sum of the products of theelements and their cofactors is 2 ð 11 C 7 ð 13, i.e.,

Problem 16 Determine the value of

Now try the following exercise

Exercise 204 Further problems on 3 by 3

(i) forming a matrix B of the cofactors of A, and(ii) transposing matrix B to give BT, where BT isthe matrix obtained by writing the rows of B

as the columns of BT Then adj A=B T

The inverse of matrix A, A1 is given by

A −1= adj A

jAj

where adj A is the adjoint of matrix A and jAj

is the determinant of matrix A

Problem 17 Determine the inverse of thematrix

(i) obtaining the matrix of the cofactors of theelements, and

(ii) transposing this matrix

The cofactor of element 3 is C 0 7



D21.The cofactor of element 4 is 2 7



D11, and

so on

The matrix of cofactors is

The transpose of the matrix of cofactors, i.e the

adjoint of the matrix, is obtained by writing the rows

Now try the following exercise

Exercise 205 Further problems on the

5 Find the inverse of

The solution of simultaneous

equations by matrices and

determinants

61.1 Solution of simultaneous

equations by matrices

(a) The procedure for solving linear simultaneous

equations in two unknowns using matrices is:

(i) write the equations in the form

a1x C b1y D c1

a2x C b2y D c2(ii) write the matrix equation corresponding

(iv) multiply each side of (ii) by the inverse

529429

329





(iv) Multiplying each side of (ii) by (iii) and membering that A ð A1 DI, the unit matrix,gives:

529429

329

29 

5729

4 ð 4  3 ð 1  19 D 0 D RHS

(b) The procedure for solving linear simultaneous

equations in three unknowns using

to these equations, i.e

d1

d2

d3(iii) determine the inverse matrix of

4332(iii) The inverse matrix of

D1 ð 14  1 ð 16 C 1 ð 5 D 35Hence the inverse of A,

A1D 135

xyzD

135

2

35(v) By comparing corresponding elements, x = 2,

y =− 3, z = 5, which can be checked in the

original equations

Now try the following exercise

Exercise 206 Further problems on solving

simultaneous equations using matrices

In Problems 1 to 5 use matrices to solve the

simultaneous equations given

6 In two closed loops of an electrical

cir-cuit, the currents flowing are given by the

7 The relationship between the

displace-ment, s, velocity, v, and acceleration, a,

of a piston is given by the equations:

3.4Rx C 7.0Px  13.2x D 11.39

6.0Rx C 4.0Px C 3.5x D 4.982.7Rx C 6.0Px C 7.1x D 15.91Use matrices to find the values of Rx, Pxand x

[Rx D 0.5, Px D 0.77, x D 1.4]

61.2 Solution of simultaneous equations by determinants

(a) When solving linear simultaneous equations in

two unknowns using determinants:

(i) write the equations in the form

a1x C b1y C c1D0

a2x C b2y C c2D0and then

(ii) the solution is given by

7x C 5y D 6.5

Following the above procedure:

143

y64.5 D

1

43 then y D 

64.5

43 D−1.5

Problem 4 The velocity of a car,

accelerating at uniform acceleration a

between two points, is given by vDu C at,

where u is its velocity when passing the first

point and t is the time taken to pass between

the two points IfvD21 m/s when t D 3.5 s

and vD33 m/s when t D 6.1 s, use

determinants to find the values of u and a,

each correct to 4 significant figures

Substituting the given values invDu C atgives:

D3.533  216.1

D12.6Similarly, DaD 1 21

Problem 5 Applying Kirchhoff’s laws to

an electric circuit results in the followingequations:

9 C j12I16 C j8I2 D5

6 C j8I1C8 C j3I2 D2 C j4

Solve the equations for I1 and I2

Following the procedure:

(i) 9 C j12I16 C j8I25 D 0

6 C j89 C j12 6 C j88 C j3

(b) When solving simultaneous equations in three

unknowns using determinants:

(i) Write the equations in the form

b1 c1 d1

b2 c2 d2

b3 c3 d3

i.e the determinant of the coefficients

obtained by covering up the x column

Dy is

a1 c1 d1

a2 c2 d2

a3 c3 d3

i.e., the determinant of the coefficients

obtained by covering up the y column

Dz is

a1 b1 d1

a2 b2 d2

a3 b2 d3

i.e the determinant of the coefficients

obtained by covering up the z column

and D is

a1 b1 c1

a2 b2 c2

a3 b3 c3

i.e the determinant of the coefficientsobtained by covering up the constantscolumn

Problem 6 A d.c circuit comprises threeclosed loops Applying Kirchhoff’s laws tothe closed loops gives the followingequations for current flow in milliamperes:2I1C3I24I3 D26

I15I23I3 D 87

7I1C2I2C6I3 D12Use determinants to solve for I1, I2 and I3

(i) Writing the equations in the

a1x C b1y C c1z C d1 D0 form gives:

2I1C3I24I326 D 0

I15I23I3C87 D 0

7I1C2I2C6I312 D 0(ii) The solution is given by

DI1 D

D3 3 87

6 12

4 5 87

2 12

C26 5 3

D3486 C 4114  2624

D−1290

DI 2 D

D236  522  412 C 609

C266  21

D 972 C 2388 C 390

D1806

I1 D 1290

129 = 10 mA,

I2 D 1806

129 = 14 mAand I3 D 1161

129 = 9 mA

Now try the following exercise

Exercise 207 Further problems on

solv-ing simultaneous equations using determinants

In problems 1 to 5 use determinants to solve

the simultaneous equations given



6 In a system of forces, the relationshipbetween two forces F1 and F2 is givenby:

5F1C3F2C6 D 03F1C5F2C18 D 0Use determinants to solve for F1and F2

[F1D1.5, F2D 4.5]

7 Applying mesh-current analysis to ana.c circuit results in the following equa-tions:

5  j4I1j4I2 D1006 0°

4 C j3  j4I2j4I1 D0Solve the equations for I1and I2, correct

i1C8i2C3i3 D 313i12i2Ci3 D 52i13i2C2i3 D6Use determinants to solve for i1, i2 and

Find the values of F1, F2 and F3 using

determinants

[F1 D2, F2 D 3, F3 D4]

10 Mesh-current analysis produces the

fol-lowing three equations:

equations using Cramers rule

Cramers rule states that if

i.e the x-column has been replaced by the R.H.S b

i.e the y-column has been replaced by the R.H.S b

i.e the z-column has been replaced by the R.H.S b

column

Problem 7 Solve the followingsimultaneous equations using Cramers rule

x C y C z D42x  3y C 4z D 333x  2y  2z D 2

(This is the same as Problem 2 and a comparison ofmethods may be made) Following the above method:

D D

D16  8  14  12

C14  9 D 14 C 16 C 5 D 35

DxD

D46  8  166  8

C166  6 D 56 C 74  60 D 70

Dy D

D166  8  44  12 C 14  99

D 74 C 64  95 D−105

DzD

Now try the following exercise

Exercise 208 Further problems on

solv-ing simultaneous equations using Cramers rule

1 Repeat problems 3, 4, 5, 7 and 8 of Exercise

206 on page 515, using Cramers rule

2 Repeat problems 3, 4, 8 and 9 of Exercise

207 on page 518, using Cramers rule

Assignment 16

This assignment covers the material

con-tained in chapters 59 to 61 The marks

for each question are shown in brackets

at the end of each question.

1 Use the laws and rules of Boolean

alge-bra to simplify the following

4 A clean room has two entrances,

each having two doors, as shown in

Fig A16.1 A warning bell must sound

if both doors A and B or doors C

and D are open at the same time

Write down the Boolean expression

depicting this occurrence, and devise a

logic network to operate the bell using

2.4I1C3.6I2C4.8I3 D1.2

3.9I1C1.3I26.5I3 D2.61.7I1C11.9I2C8.5I3 D0Using matrices, solve the equations for I1, I2

Multiple choice questions on

5 For the curve shown in Figure M4.1, which of

the following statements is incorrect?

(a) P is a turning point

Figure M4.1

7 If y D 5px32, d y

d x is equal to:

(a) 152

p

(c) 52

4e

(c) e2x

4 2x  1 C c (d) 2e

2xx 2 C c

9 An alternating current is given by

i D 4 sin 150t amperes, where t is the time

in seconds The rate of change of current at

t D0.025 s is:

10 A vehicle has a velocityvD2 C 3t m/s after

tseconds The distance travelled is equal to thearea under the v/t graph In the first 3 secondsthe vehicle has travelled:

(a) 11 m (b) 33 m (c) 13.5 m (d) 19.5 m

11 Differentiating y D p1

x C2 with respect to xgives:

(a) p1

2px3(c) 2  1

x3

12 The area, in square units, enclosed by the curve

y D2x C 3, the x-axis and ordinates x D 1 and

xC2317

22 Given ft D 3t42, f0tis equal to:

t D0.1 s, the rate of change of current is:

25

 3 2

27

5 sin 3t  3 cos 5t d t is equal to:

(a) 5 cos 3t C 3 sin 5t C c

29 The velocity of a car (in m/s) is related to time

tseconds by the equationvD4.5C18t4.5t2

The maximum speed of the car, in km/h, is:

p

x33x C c

31 An alternating voltage is given by

vD10 sin 300t volts, where t is the time in

seconds The rate of change of voltage when

t D0.01 s is:

32 The r.m.s value of y D x2between x D 1 and

x D3, correct to 2 decimal places, is:

35 The equation of a curve is y D 2x36x C 1.The minimum value of the curve is:

36 The volume of the solid of revolution whenthe curve y D 2x is rotated one revolutionabout the x-axis between the limits x D 0 and

x(c) 6  1

1

x239

4is:

43 An alternating current, i amperes, is given by

i D 100 sin 2ft amperes, where f is the

frequency in hertz and t is the time in seconds

The rate of change of current when t D 12 ms

and f D 50 Hz is:

44 A metal template is bounded by the curve

y D x2, the x-axis and ordinates x D 0 and

x D2 The x-co-ordinate of the centroid of the

46 The area under a force/distance graph gives the

work done The shaded area shown between p

and q in Figure M4.2 is:

Answers to multiple choice questions

Multiple choice questions on

Chord 139 Circle 139 equation of 143, 267 Circumference 139 Class 312

interval 312 Coefficient of correlation 347 proportionality 42

Combination of waveforms 287 Combinational logic networks 497 Combinations 112, 332

Common difference 106 logarithms 89 ratio 109 Completing the square 82 Complex conjugate 294 equations 295 numbers 291 applications of 299 powers of 303 roots of 304 waveforms 192 Compound angles 214 Computer numbering systems 16 Cone 145

Confidence, coefficient 360 intervals 360

Continuous data 307 functions 273 Conversion of a sin ωt C b cos ωt into R sinωt C ˛ 216 tables and charts 28

Co-ordinates 231 Correlation 347 Cosecant 172 Cosine 172 rule 198, 289 wave production 185 Cotangent 172

Couple 491 Cramer’s rule 520 Cubic equations 264, 266 Cuboid 145

Cumulative frequency distribution 313, 316 Cylinder 145

Factorization 38, 80 Factor theorem 46 False axes 238 Finite discontinuities 273 First moment of area 466, 475 Formula 30

quadratic 84 Formulae, transposition of 74 Fractional form of trigonometric ratios 174 Fractions 1

partial 51 Frequency 190, 307 distribution 312, 315 polygon 313, 315 Frustum of pyramids and cones 151 sphere 155

Functional notation 375, 377 Function of a function rule 389 Functions and their curves 266 Fundamental 192

Geometric progression 109 Gradient of a curve 376 straight line graph 231 Graphical solution of equations 258 Graphs 230

of cubic equations 264 exponential functions 98 linear and quadratic equations simultaneously 263 logarithmic functions 93

quadratic equations 57 simultaneous equations 258 straight lines 231

trigonometric functions 182, 185 Graphs with logarithmic scales 251 Grouped data 312

Harmonic analysis 192 Harmonics 192 H.C.F 36 Heptagon 131 Hexadecimal numbers 20 Hexagon 131

Histogram 313, 316, 321

of probability 335, 337 Hooke’s law 42

Horizontal bar chart 308 Hyperbola 267

rectangular 268 Hyperbolic logarithms 89, 100

Identity 57 trigonometric 208 Imaginary number 291 Improper fraction 1

Log-linear graph paper 254

Log-log graph paper 251

Logic circuits 495

universal 500

Mantissa 13 Matrices 504

to solve simultaneous equations 514 –516 Matrix 504

adjoint 511 determinant of 508, 510 inverse 509, 511 reciprocal 509 unit 508 Maximum value 259, 396 and minimum problems 399 Mean value of waveform 164 Mean values 319, 320, 457 Measures of central tendency 319 Median 319, 320

Mensuration 131 Mid-ordinate rule 161, 441, 451 Minimum value 259, 396 Mixed number 1 Modal value 319, 320 Modulus 296 Multiple-choice questions 127, 224, 369, 522

Nand-gate 495 Napierian logarithms 89, 100 Natural logarithms 89, 100 Newton –Raphson method 123 Non-terminating decimal 5 Nor-gate 496

Normal curve 340 distribution 340 equations 351 probability paper 344 standard variate 340 Normals 403

Nose-to-tail method 282 Not-function 484 Not-gate 495 Number sequences 106 Numerator 1

Numerical integration 439

Octagon 131 Octal 18 Odd function 273 Ogive 313, 316 Ohm’s law 42 Ordinate 231 Or-function 483 Or-gate 495

Pappus’ theorem 471 Parabola 259 Parallel-axis theorem 475 Parallelogram 131 method 282

Practical problems, binomial theorem 120

maximum and minimum 399

Radians 140, 190 Radius of gyration 475 Radix 16

Rates of change 392 Ratio and proportion 3 Real part of complex number 291 Reciprocal 9

Reciprocal matrix 509 Rectangular axes 231 co-ordinates 197 hyperbola 268 prism 145 Rectangle 131 Reduction of non-linear to linear form 243 Regression 351

coefficients 351 Relative frequency 307 Remainder theorem 48 Resolution of vectors 283 Resultant phasor 288 Rhombus 131 Root mean square value 459 Root of equation 80 complex number 304 Rounding off errors 24

Sample data 307, 356 Sampling distributions 356 Scalar quantity 281 Scatter diagram 347 Secant 172

Second moments of area 475 Sector 139

Segment 139 Semi-interquartile range 324 Set 307

Significant figures 5 Simple equations 57 practical problems 61 Simpson’s rule 161, 443, 451 Simultaneous equations 65

by Cramer’s rule 520

by determinants 516 –520

by matrices 514 –516 graphically 258 practical problems 70 Sine 172

rule 198, 289 Sine wave 164, 185 production 185 Sinusoidal form A sinωt š ˛ 189 Slope 231

Small changes 404