Solutions to Exercises

1. Introduction

Perform the operation 3*4+6. The order of the operations will be discussed in subsequent chapters.

>> 3*4+6

ans =

18

Perform the operation cos(5). The value of 5 is in radians.

>> cos (5)

ans =

0.2837

Perform the operation i_Image1

for x = 4.

>> x = 4

x =

4

>> 3*sqrt(6+x)

ans =

9.4868

Assign the value of 5.2 to the variable y.

>> y = 5.2

y =

5.2000

Assign the values of 3 and 4 to the variables x and y, respectively, then calculate the value of z where z = 2x − 7y.

>> x = 3;
>> y = 4;
>> z = 2*x - 7*y

z =

-22

Obtain help on the inv command.

>> help inv
INV Matrix inverse.
INV(X) is the inverse of the square matrix X.
A warning message is printed if X is badly scaled or nearly singular.

See also slash, pinv, cond, condest, lsqnonneg, lscov.

Overloaded functions or methods (ones with the same name in other directories) help sym/inv.m

Reference page in Help browser doc inv

Generate the following matrix: A

>> A = [1 0 2 -3 ; 0 5 2 2 ; 1 2 3 4 ; -2 0 1 3]

A =

1 0 2 -3
0 5 2 2
1 2 3 4
-2 0 1 3

Generate the following vector b

>> b = [1 ; 2 ; 3 ; 4]

b =

1
2
3
4

Evaluate the vector c where where A is the matrix given in Exercise 7 above and b is the vector given in Exercise 8 above.

>> c = A*b

c =

-5
24
30
13

Solve the following system of simultaneous algebraic equation using Gaussian elimination.

>> A = [ 5 2 ; 1 3]

A =

5 2
1 3

>> b = [3 ; −1]

b =

3
-1

>> x = A\b

x =

0.8462
-0.6154

Solve the system of simultaneous algebraic equations of Exercise 10 above using matrix inversion.

>> x = inv(A)*b

x =

0.8462
−0.6154

Generate the following matrix X:

>> X = [1 0 6 ; 1 2 3 ; 4 5 -2]

X =

1 0 6
1 2 3
4 5 -2

Extract the sub-matrix in rows 2 to 3 and columns 1 to 2 of the matrix X in Exercise 12 above.

>> X(2:3,1:2)

ans =

1 2
4 5

Extract the second column of the matrix X in Exercise 12 above.

>> X(1:3,2)

ans =
0
2
5

Extract the first row of the matrix X in Exercise 12 above.

>> X(1,1:3)

ans =

1 0 6

Extract the element in row 1 and column 3 of the matrix X in Exercise 12 above.

>> X(1,3)

ans =

6

Generate the row vector x with integer values ranging from 1 to 9.

>> x = [1 2 3 4 5 6 7 8 9]

x =

1 2 3 4 5 6 7 8 9

Plot the graph of the function y = x³ − 2 for the range of the values of x in Exercise 17 above.

>> y = x.^3 - 2

y =

-1 6 25 62 123 214 341 510 727

>> plot(x,y)

Generate a 4 x 4 magic square. What is the total of each row, column, and diagonal in this matrix.

>> magic(4)

ans =

16 2 3 13
5 11 10 8
9 7 6 12
4 14 15 1

The sum of each row, column, or diagonal is 34.

2. Arithmetic Operations

Perform the addition operation 7+9.

>> 7+9

ans =

16

Perform the subtraction operation 16-10.

>> 16−10

ans =

6

Perform the multiplication operation 2*9.

>> 2*9

ans =

18

Perform the division operation 12/3.

>> 12/3

ans =

4

Perform the division operation 12/5.

>> 12/5

ans =

2.4000

Perform the exponentiation operation 3^5.

>> 3^5

ans =

243

Perform the exponentiation operation 3*(-5).

>> 3^(-5)

ans =

0.0041

Perform the exponentiation operation (-3)^5.

>> (−3)^5

ans =

−243

Perform the exponentiation operation -3^5.

>> −3^5

ans =

−243

Compute the value of

>> 2*pi/3

ans =

2.0944

Obtain the value of the smallest number that can be handled by MATLAB.

>> eps

ans =

2.2204e-016

Perform the multiple operations 5+7-15.

>> 5+7-15

ans =

-3

Perform the multiple operations (6*7)+4.

>> 6*7 +4

ans =

46

Perform the multiple operations 6*(7+4).

>> 6*(7+4)

ans =

66

Perform the multiple operations 4.5 + (15/2).

>> 4.5 + 15/2

ans =

12

Perform the multiple operations (4.5 + 15)/2.

>> (4.5 + 15)/2

ans =

9.7500

Perform the multiple operations (15 – 4 + 12)/5 – 2*(7^4)/100.

>> (15–4+12) /5 – 2*(7^4)/100

ans =

-43.4200

Perform the multiple operations (15 – 4) + 12/5 – (2*7)^4/100.

>> (15–4) + 12/5 – (2*7)^4/100

ans =

−370.7600

Define the number 2/3 as a symbolic number.

>> sym(2/3)

ans =

2/3

Perform the fraction addition (2/3) + (3/4) numerically.

>> 2/3 + 3/4

ans =

1.4167

Perform the fraction addition (2/3) + (3/4) symbolically.

>> sym((2/3)+(3/4))

ans =

17/12

3. Variables

Perform the operation 2*3+7 and store the result in the variable w.

>> 2*3+7

ans =

13

Define the three variables a, b, and c equal to 4, −10, and 3.2, respectively.

>> a = 4

a =

4

>> b = −10

b =

−10

>> c = 3.2

c =

3.2000

Define the two variables y and Y equal to 10 and 100. Are the two variables identical?

>> y = 10

y =

10

>> Y = 100

Y =

100

The two variables are not identical.

Let x = 5.5 and y = −2.6. Calculate the value of the variable z = 2x−3y.

x =

5.5000

>> y = −2.6

y =

−2.6000

>> z = 2*x − 3*y

z =

18.8000

In Exercise 4 above, calculate the value of the variable w = 3y − z + x/y.

>> w = 3*y − z + x/y

w =

−28.7154

Let r = 6.3 and s = 5.8. Calculate the value of the variable final defined by final = r + s - r*s.

>> r = 6.3

r =

6.3000

>> s = 5.8

s =

5.8000

>> final = r + s - r*s

final =

-24.4400

In Exercise 6 above, calculate the value of the variable this_is_the_result defined by this_is_the_result = r^2 – s^2.

>> this_is_the_result = r^2 − s^2

this_is_the_result =

6.0500

Define the three variable width, Width, and WIDTH equal to 1.5, 2.0, and 4.5, respectively. Are these three variables identical?

>> width = 1.5

width =

1.5000

>> Width = 2.0

Width =

2

>> WIDTH = 4.5

WIDTH =

4.5000

The three variables are not identical.

Write the following comment in MATLAB: This line will not be executed.

% This line will not be executed.

Assign the value of 3.5 to the variable s then add a comment about this assignment on the same line.

>> s = 3.5 % the variable s is assigned the value 3.5

s =

3.5000

Define the values of the variables y1 and y2 equal to 7 and 9 then perform the calculation y3 = y1 – y2/3. (Note: 2 in the formula is a subscript and should not be divided by 3).

>> y1 = 7

y1 =

7

>> y2 = 9

y2 =

9

>> y3 = y1 − y2/3

y3 =

4

Perform the operation 2*m – 5. Do you get an error? Why?

>> 2*m − 5
??? Undefined function or variable 'm'.

We get an error message because the variable m is not defined.

Define the variables cost and profit equal to 175 and 25, respectively, then calculate the variable sale_price defined by sale_price = cost + profit.

>> cost = 175

cost =

175

>> profit = 25

profit =

25

>> sale_price = cost + profit

sale_price =

200

Define the variable centigrade equal to 28 then calculate the variable fahrenheit defined by fahrenheit = (centigrade*9/5) + 32.

>> centigrade = 28

centigrade =

28

>> fahrenheit = (centigrade*9/5) + 32

fahrenheit =

82.4000

Use the format short and format long commands to write the values of 14/9 to four decimals and sixteen digits, respectively.

>> format long
>> 14/9

ans =

1.55555555555556

>> format short
>> 14/9

ans =

1.5556

Perform the who command to get a list of the variables stored in this session.

>> who

Your variables are:

WIDTH fahrenheit x Width final y Y profit y1 a r y2 ans s y3 b sale_price z c this_is_the_result centigrade w cost width

Perform the whos command to get a list of the variables stored in this session along with their details.

>> whos

See Table
Clear all the variables stored in this session by using the clear command.

>> clear
>>
Calculate both the area and perimeter of a rectangle of sides 5 and 7. No units are used in this exercise.

a =

5

>> b = 7

b =

7

>> area = a*b

area =

35

>> perimeter = 2*(a+b)

perimeter =

24
Calculate both the area and perimeter of a circle of radius 6.45. No units are used in this exercise.

>> r = 6.45

r =

6.4500

>> area = pi*r^2

area =

130.6981

>> perimeter = 2*pi*r

perimeter =

40.5265

Define the symbolic variables x and z with values 4/5 and 14/17.

>> x = 4/5

x =

0.8000

>> sym(x)

ans =

4/5

>> z = 14/17

z =

0.8235

>> sym(z)

ans =

14/17

In Exercise 21 above, calculate symbolically the value of the symbolic variable y defined by y = 2x – z.

>> y = sym(2*x − z)

y =

66/85

Calculate symbolically the area of a circle of radius 2/3 without obtaining a numerical value. No units are used in this exercise.

>> radius1 = 2/3

radius1 =

0.6667

>> radius1 = sym(radius1)

radius1 =

2/3

>> area = pi*radius1^2

area =

4/9*pi

Calculate symbolically the volume of a sphere of radius 2/3 without obtaining a numerical value. No units are used in this exercise.

>> radius2 = 2/3

radius2 =

0.6667

>> radius2 = sym(radius2)

radius2 =

2/3

>> volume = (4/3)*pi*radius2^3

volume =

32/81*pi

In Exercise 23 above, use the double command to obtain the numerical value of the answer.

>> double(area)

ans =

1.3963

In Exercise 24 above, use the double command to obtain the numerical value of the answer.

>> double(volume)

ans =

1.2411

Define the symbolic variables y and date without assigning any numerical values to them.

>> y = sym('y')

y =

y

>> date = sym('date')

date =

date

4. Mathematical Functions

Compute the square root of 10.

>> sqrt(10)

ans =

3.1623

Compute the factorial of 7.

>> factorial(7)

ans =

5040

Compute the cosine of the angle 45 where 45 is in radians.

>> cos(45)

ans =

0.5253

Compute the cosine of the angle 45 where 45 is in degrees.

>> cos(45*pi/180)

ans =

0.7071

Compute the sine of the angle of 45 where 45 is in degrees.

>> sin(45*pi/180)

ans =

0.7071

Compute the tangent of the angle 45 where 45 is in degrees.

>> tan(45*pi/180)

ans =

1.0000

Compute the inverse tangent of 1.5.

>> atan(1.5)

ans =

0.9828

The above result is in radians.

Compute the tangent of the angle . Do you get an error? Why?

>> tan(3*pi/2)

ans =

5.4437e+015

We get a very large number approaching infinity because the tanget function is not define at .

Compute the value of exponential function e³.

>> exp(3)

ans =

20.0855

Compute the value of the natural logarithm ln 3.5.

>> log(3.5)

ans =

1.2528

Compute the value of the logarithm log₁₀ 3.5.

>> log10(3.5)

ans =

0.5441

Use the MATLAB rounding function round to round the value of 2.43.

>> round(2.43)

ans =

2

Use the MATLAB remainder function rem to obtain the remainder when dividing 5 by 4.

>> rem(5,4)

ans =

1

Compute the absolute value of -3.6.

>> abs(-3.6)

ans =

3.6000

Compute the value of the expression

>> 1.5 - 2*sqrt(6.7/5)

ans =

-0.8152

Compute the value of sin ² π + cos² π.

>> sin(pi)^2 + cos(pi)^2

ans =

1

Compute the value of log₁₀ 0. Do you get an error? Why?

>> log10(0)
Warning: Log of zero.
> In log10 at 17

ans =

-Inf

We get minus infinity because the logarithmic function is not defined at zero.

Let

and y = 2π. Compute the value of the expression 2sin x cos y.

>> x = 3*pi/2

x =

4.7124

>> y = 2*pi

y =

6.2832

>> 2*sin(x)*cos(y)

ans =

-2

Compute the value of symbolically and simplify the result.

>> sym(sqrt(45))

ans =

sqrt(45)

>> simplify(ans)

ans =

3*5^(1/2)

Compute the value of

numerically.

>> double(ans)

ans =

6.7082

Compute the sine of the angle 45 (degrees) symbolically.

>> sym(sin(45*pi/180))

ans =

sqrt(1/2)

Compute the cosine of the angle 45 (degrees) symbolically.

> sym(cos(45*pi/180))

ans =

sqrt(1/2)

Compute the tangent of the angle 45 (degrees) symbolically.

>> sym(tan(45*pi/180))

ans =

1

Compute the value of e^π/2 symbolically.

>> sym(exp(pi/2))

ans =

5416116035097439*2^(-50)

Compute the value of e^π/2 numerically.

>> double(ans)

ans =

4.8105

5. Complex Numbers

Compute the square root of -5.

>> sqrt(-5)

ans =

0 + 2.2361i

Define the complex number i_Image1

>> 4-3*sqrt(-8)

ans =

4.0000 - 8.4853i

Define the two complex numbers with variables x and y where x = 2 − 6i and y = 4 +11i.

>> x = 2-6i

x =

2.0000 - 6.0000i

>> y=4+11i

y =

4.0000 +11.0000i

In Exercise 3 above, perform the addition and subtraction operations x + y and x − y.

>> x+y

ans =

6.0000 + 5.0000i

>> x-y

ans =

-2.0000 -17.0000i

In Exercise 3 above, perform the multiplication and division operations x y and

>> x*y

ans =

74.0000 - 2.0000i

>> x/y

ans =

-0.4234 - 0.3358i

In Exercise 3 above, perform the exponentiation operations x⁴ and y ^-3.

>> x^4

ans =

4.4800e+002 +1.5360e+003i

>> y^(-3)

ans =

-5.3979e-004 +3.1229e-004i

In Exercise 3 above, perform the multiple operations 4x − 3y + 9.

>> 4*x-3*y+9

ans =

5.0000 -57.0000i

In Exercise 3 above, perform the multiple operations ix − 2y − 1.

>> i*x-2*y-1

ans =

-3.0000 -20.0000i

Compute the magnitude of the complex number 3 − 5i.

>> abs(3−5i)

ans =

5.8310

Compute the angle of the complex number 3 − 5i in radians.

>> angle(3−5i)

ans =

-1.0304

Compute the angle of the complex number 3 − 5i in degrees.

>> angle(3-5i)*180/pi

ans =

-59.0362

Extract the real and imaginary parts of the complex number 3 − 5i.

>> real(3−5i)

ans =

3

>> imag(3−5i)

ans =

−5

Obtain the complex conjugate of the complex number 3 − 5i.

>> conj(3-5i)

ans =

3.0000 + 5.0000i

Compute the sine, cosine, and tangent functions of the complex number 3 − 5i.

>> sin(3-5i)

ans =

10.4725 +73.4606i

>> cos(3-5i)

ans =

-73.4673 +10.4716i

>> tan(3-5i)

ans =

-0.0000 - 0.9999i

Compute e^3-5i and ln(3 − 5i).

>> exp(3-5i)

ans =

5.6975 +19.2605i

>> log(3-5i)

ans =

1.7632 - 1.0304i

Compute the values of sin and e^{πi /2}.

>> sin(pi*i/2)

ans =

0 + 2.3013i

>> cos(pi*i/2)

ans =

2.5092

>> exp(pi*i/2)

ans =

0.0000 + 1.0000i

Compute the value of (3 + 4i)^(2-i).

>> (3+4i)^(2-i)

ans =

61.3022 +15.3369i

Obtain

symbolically.

>> sym(sqrt(-13))

ans =

(0)+(sqrt(13))*i

Obtain the magnitude of the complex number 3− 5i symbolically.

>> sym(abs(3-5i))

ans =

sqrt(34)

Obtain the angle of the complex number 3− 5i symbolically. Make sure that you use the double command at the end.

>> sym(angle(3-5i))

ans =

-4640404691986088*2^(-52)

>> double(ans)

ans =

-1.0304

Obtain the cosine function of the complex number 3− 5i symbolically. Make sure that you use the double command at the end.

>> sym(cos(3-5i))

ans =

(-5169801091137321*2^(-46))+(5894962905280379*2^(-49))*i

>> double(ans)

ans =

-73.4673 +10.4716i

6. Vectors

Store the vector [2 4 -6 0] in the variable w.

>> w = [2 4 -6 0]

w =

2 4 -6 0

In Exercise 1 above, extract the second element of the vector w.

> w(2)

ans =

4

In Exercise 1 above, generate the vector z where i_Image1

>> z = pi*w/2

z =

3.1416 6.2832 -9.4248 0

In Exercise 3 above, extract the fourth element of the vector z.

>> z(4)

ans =

0

In Exercise 3 above, extract the first three elements of the vector z.

>> z(1:3)

ans =

3.1416 6.2832 -9.4248

In Exercise 3 above, find the length of the vector z.

>> length(z)

ans =

4

In Exercise 3 above, find the total sum of the values of the elements of the vector z.

>> sum(z)

ans =

0

In Exercise 3 above, find the minimum and maximum values of the elements of the vector z.

>> min(z)

ans =

-9.4248

>> max(z)

ans =

6.2832

Generate a vector r with real values between 1 and 10 with an increment of 2.5.

>> r = (1:2.5:10)

r =

1.0000 3.5000 6.0000 8.5000

Generate a vector s with real values of ten numbers that are equally spaced between 1 and 100.

>> s = linspace(1,100,10)

s =

1 12 23 34 45 56 67 78 89 100

Form a new vector by joining the two vectors [9 3 -2 5 0] and [1 2 - 4].

>> a = [9 3 -2 5 0]

a =

9 3 -2 5 0

>> b = [1 2 -4]

b =

1 2 -4

>> c = [a b]

c =

9 3 -2 5 0 1 2 -4

Form a new vector by joining the vector [9 3 -2 5 0] with the number 4.

>> d = [a 4]

d =

9 3 -2 5 0 4

Add the two vectors [0.2 1.3 -3.5] and [0.5 -2.5 1.0].

>> x = [0.2 1.3 -3.5]

x =

0.2000 1.3000 -3.5000

>> y = [0.5 -2.5 1.0]

y =

0.5000 -2.5000 1.0000

>> x+y

ans =

0.7000 -1.2000 -2.5000

Subtract the two vectors in Exercise 13 above.

>> x-y

ans =

-0.3000 3.8000 -4.5000

Try to multiply the two vectors in Exercise 13 above. Do you get an error message? Why?

>> x*y
??? Error using ==> mtimes

Inner matrix dimensions must agree.

We get an error because the two vectors do not have the same length.

Multiply the two elements in Exercise 13 above element by element.

>> x.*y

ans =

0.1000 -3.2500 -3.5000

Divide the two elements in Exercise 13 above element by element.

>> x./y

ans =

0.4000 -0.5200 -3.5000

Find the dot product of the two vectors in Exercise 13 above.

>> x*y'

ans =

-6.6500

Try to add the two vectors [1 3 5] and [3 6]. Do you get an error message? Why?

>> r = [ 1 3 5 ]

r =

1 3 5

>> s = [ 3 6 ]

s =

3 6

>> r+s
??? Error using ==> plus
Matrix dimensions must agree.

We get an error message because the two vectors do not have the same length.

Try to subtract the two vectors in Exercise 20 above. Do you get an error message? Why?

>> r-s
??? Error using ==> minus
Matrix dimensions must agree.

We get an error message because the two vectors do not have the same length.

Let the vector w be defined by w = [0.1 1.3 -2.4]. Perform the operation of scalar addition 5+w.

>> w = [0.1 1.3 -2.4]

w =

0.1000 1.3000 -2.4000

>> 5+w

ans =

5.1000 6.3000 2.6000

In Exercise 22 above, perform the operation of scalar subtraction -2-w.

>> 2-w

ans =

1.9000 0.7000 4.4000

In Exercise 22 above, perform the operation of scalar multiplication 1.5*w.

>> 1.5*w

ans =

0.1500 1.9500 -3.6000

In Exercise 22 above, perform the operation of scalar division w/10.

>> w/10

ans =

0.0100 0.1300 -0.2400

In Exercise 22 above, perform the operation 3 − 2*w/5.

>> 3-2*w/5

ans =

2.9600 2.4800 3.9600

Define the vector b by b = [0 pi/3 2pi/3 pi]. Evaluate the three vectors sin b, cosb, and tan b (element by element).

>> b = [0 pi/3 2*pi/3 pi]

b =

0 1.0472 2.0944 3.1416

>> sin(b)

ans =

0 0.8660 0.8660 0.0000

>> cos(b)

ans =

1.0000 0.5000 -0.5000 -1.0000

>> tan(b)

ans =

0 1.7321 -1.7321 -0.0000

In Exercise 26 above, evaluate the vector e^b (element by element).

>> exp(b)

ans =

1.0000 2.8497 8.1205 23.1407

In Exercise 26 above, evaluate the vector

(element by element).

>> sqrt(b)

ans =

0 1.0233 1.4472 1.7725

Try to evaluate the vector 3^b. Do you get an error message? Why?

>> 3^b
??? Error using ==> mpower
Matrix must be square.

Yes, because this operation needs to be performed element by element.

Perform the operation in Exercise 29 above element by element?

>> 3.^b

ans =

1.0000 3.1597 9.9834 31.5443

Generate a vector of 1’s with a length of 4 elements.

> ones(1,4)

ans =

1 1 1 1

Generate a vector of 0’s with a length of 6 elements.

>> zeros(1,6)

ans =

0 0 0 0 0 0

Sort the elements of the vector [0.35 -1.0 0.24 1.30 -0.03] in ascending order.

>> k = [0.35 -1.0 0.24 1.30 -0.03]

k =

0.3500 -1.0000 0.2400 1.3000 -0.0300

>> sort(k)

ans =

-1.0000 -0.0300 0.2400 0.3500 1.3000

Generate a random permutation vector with 5 elements.

>> randperm(5)

ans =

2 4 3 5 1

For the vector [2 4 -3 0 1 5 7], determine the range, mean, and median.

>> v = [2 4 -3 0 1 5 7]

v =

2 4 -3 0 1 5 7

>> range(v)

ans =

10

>> mean(v)

ans =

2.2857

>> median(v)

ans =

2

Define the symbolic vector x = [r s t u v].

>> syms r s t u v
>> x = [r s t u v]

x = [ r, s, t, u, v]

In Exercise 36 above, perform the addition operation of the two vectors x and [1 0 -2 3 5] to obtain the new symbolic vector y.

>> r = [1 0 -2 3 5]

r =

1 0 -2 3 5

>> y = x+r

y =

[ r+1, s, t-2, u+3, v+5]

In Exercise 37 above, extract the third element of the vector y.

>> y(3)

ans =

t-2

In Exercise 37 above, perform the operation 2*x/7 + 3*y.

>> 2*x/7 + 3*y

ans =

[ 23/7*r+3, 23/7*s, 23/7*t-6, 23/7*u+9, 23/7*v+15]

In Exercise 37 above, perform the dot products x*y′ and y*x′. Are the two results the same? Why?

>> x*y'

ans =

r*(1+conj(r))+s*conj(s)+t*(-2+conj(t))+u*(3+conj(u))+v*(5+conj(v))

>> y*x'

ans =

(r+1)*conj(r)+s*conj(s)+(t-2)*conj(t)+(u+3)*conj(u)+(v+5)*conj(v)

The two results are different because MATLAB treats these symbolic variables as complex variables by default. If they were real variables, then the results would be the same.

In Exercise 37 above, find the square root of the symbolic vector x+y.

>> sqrt(x+y)

ans =

[ (2*r+1)^(1/2), 2^(1/2)*s^(1/2), (2*t-2)^(1/2), (2*u+3)^(1/2), (2*v+5)^(1/2)]

7. Matrices

Generate the following rectangular matrix in MATLAB. What is the size of this matrix?

>> A = [3 0 -2 ; 1 3 5]

A =

3 0 -2
1 3 5

A is a rectangular matrix of size 2x3.

In Exercise 1 above, extract the element in the second row and second column of the matrix A.

>> A(2,2)

ans =

3

In Exercise 1 above, generate a new matrix B of the same size by multiplying the matrix A by the number

>> B = 3*pi*A/2

B =

14.1372 0 -9.4248
4.7124 14.1372 23.5619

In Exercise 3 above, extract the element in the first row and third column of the matrix B.

>> B(1,3)

ans =

-9.4248

In Exercise 3 above, extract the sub-matrix of the elements in common between the first and second rows and the second and third columns of the matrix B. What is the size of this new submatrix?

>> B(1:2,2:3)

ans =

0 -9.4248
14.1372 23.5619

The above sub-matrix has size 2x2.

In Exercise 3 above, determine the size of the matrix B using the MATLAB command size?

>> size(B)

ans =

2 3

In Exercise 3 above, determine the largest of the number of rows and columns of the matrix B using the MATLAB command length?

>> length(B)

ans =

3

In Exercise 3 above, determine the number of elements in the matrix B using the MATLAB command numel?

>> numel(B)

ans =

6

In Exercise 3 above, determine the total sum of each column of the matrix B? Determine also the minimum value and the maximum value of each column of the matrix B?

>> sum(B)

ans =

18.8496 14.1372 14.1372

>> min(B)

ans =

4.7124 0 -9.4248

>> max(B)

ans =

14.1372 14.1372 23.5619

Combine the three vectors [1 3 0 -4], [5 3 1 0], and [2 2 -1 1] to obtain a new matrix of size 3x4.

>> a = [1 3 0 -4]

a =

1 3 0 -4

>> b = [5 3 1 0]

b =

5 3 1 0

>> c = [2 2 -1 1]

c =

2 2 -1 1

>> d = [a ; b ; c]

d =

1 3 0 -4
5 3 1 0
2 2 -1 1

Perform the operations of matrix addition and matrix subtraction on the following two matrices:

>> R = [1 2 0 ; 7 5 -3 ; 3 1 1]

R =

1 2 0
7 5 -3
3 1 1

>> S = [1 3 -2 ; 3 5 7 ; 2 3 0]

S =

1 3 -2
3 5 7
2 3 0

>> R+S

ans =

2 5 -2
10 10 4
5 4 1

>> R-S

ans =

0 -1 2
4 0 -10
1 -2 1

In Exercise 11 above, multiply the two matrices R and S element-by-element.

>> R.*S

ans =

1 6 0
21 25 -21
6 3 0

In Exercise 11 above, divide the two matrices R and S element-byelement.

>> R./S
Warning: Divide by zero.

ans =

1.0000 0.6667 0
2.3333 1.0000 -0.4286
1.5000 0.3333 Inf

Notice the division by zero (not allowed) above in the last element.

In Exercise 11 above, perform the operation of matrix multiplication on the two matrices R and S. Do you get an error? Why?

>> R*S

ans =

7 13 12
16 37 21
8 17 1

We do not get an error because the number of columns of the first matrix is equal to the number of rows of the second matrix.

Add the number 5 to each element of the matrix X given below:

>> X = [1 -2 0 1 ; 2 3 6 2 ; -3 5 2 1 ; 5 -2 4 4]

X =

1 -2 0 1
2 3 6 2
-3 5 2 1
5 -2 4 4

>> 5+X

ans =

6 3 5 6
7 8 11 7
2 10 7 6
10 3 9 9

In Exercise 15 above, subtract the number 3 from each element of the matrix X.

>> X-3

ans =

-2 -5 -3 -2
-1 0 3 -1
-6 2 -1 -2
2 -5 1 1

In Exercise 15 above, multiply each element of the matrix X by the number -3.

>> -3*X

ans =

-3 6 0 -3
-6 -9 -18 -6
9 -15 -6 -3
-15 6 -12 -12

In Exercise 15 above, divide each element of the matrix X by the number 2.

>> X/2

ans =

0.5000 -1.0000 0 0.5000
1.0000 1.5000 3.0000 1.0000
-1.5000 2.5000 1.0000 0.5000
2.5000 -1.0000 2.0000 2.0000

In Exercise 15 above, perform the following multiple scalar operation –3* X / 2.4 + 5.5.

>> -3*X/2.4 + 5.5

ans =

4.2500 8.0000 5.5000 4.2500
3.0000 1.7500 -2.0000 3.0000
9.2500 -0.7500 3.0000 4.2500
-0.7500 8.0000 0.5000 0.5000

Determine the sine, cosine, and tangent of the matrix B given below (element-by-element).

>> B = [pi/3 2*pi/3 ; 2*pi/3 pi]

B =

1.0472 2.0944
2.0944 3.1416

>> sin(B)

ans =

0.8660 0.8660
0.8660 0.0000

>> cos(B)

ans =

0.5000 -0.5000
-0.5000 -1.0000

>> tan(B)

ans =

1.7321 -1.7321
-1.7321 -0.0000

In Exercise 20 above, determine the square root of the matrix B element-by-element.

>> sqrt(B)

ans =

1.0233 1.4472
1.4472 1.7725

In Exercise 20 above, determine the true square root of the matrix B.

>> sqrtm(B)

ans =

0.5821 + 0.3598i 0.9419 - 0.2224i
0.9419 - 0.2224i 1.5240 + 0.1374i

Notice that we obtained a complex matrix. Try to check this answer by multiplying this complex matrix by itself to get the original matrix.

In Exercise 20 above, determine the exponential of the matrix B element-by-element.

>> exp(B)

ans =

2.8497 8.1205
8.1205 23.1407

In Exercise 20 above, determine the true exponential of the matrix B.

>> expm(B)

ans =

23.9028 37.4119
37.4119 61.3147

In Exercise 20 above, determine the natural logarithm of the matrix B element-by-element.

>> log(B)

ans =

0.0461 0.7393
0.7393 1.1447

In Exercise 20 above, determine the true natural logarithm of the matrix B.

>> logm(B)
Warning: Principal matrix logarithm is not defined for A with
nonpositive real eigenvalues. A non-principal matrix
logarithm is returned.
> In funm at 153
In logm at 27

ans =

-0.5995 + 2.2733i 1.2912 - 1.4050i
1.2912 - 1.4050i 0.6917 + 0.8683i
Notice that we obtain a non-principal complex matrix.

In Exercise 20 above, perform the exponential operation 4^B.

>> 4^B

ans =

130.0124 209.2159
209.2159 339.2283

In Exercise 27 above, repeat the same exponential operation but this time element-by-element.

>> 4.^B

ans =

4.2705 18.2369
18.2369 77.8802

In Exercise 20 above, perform the operation B⁴.

>> B^4

ans =

107.0297 173.1717
173.1717 280.2015

Generate a rectangular matrix of 1’s of size 2x3.

>> ones(2,3)

ans =

1 1 1
1 1 1

Generate a rectangular matrix of 0’s of size 2x3.

>> zeros(2,3)

ans =

0 0 0
0 0 0

Generate a rectangular identity matrix of size 2x3.

>> eye(2,3)

ans =

1 0 0
0 1 0

Generate a square matrix of 1’s of size 4.

>> ones(4)

ans =

1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1

Generate a square matrix of 0’s of size 4.

>> zeros(4)

ans =

0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0

Generate a square identity matrix of size 4.

>> eye(4)

ans =

1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

Determine the transpose of the following matrix:

>> C = [1 2 -3 0 ; 2 5 2 -3 ; 1 3 7 -2 ; 2 3 -1 3]

C =

1 2 -3 0
2 5 2 -3
1 3 7 -2
2 3 -1 3

>> C'

ans =

1 2 1 2
2 5 3 3
-3 2 7 -1
0 -3 -2 3

In Exercise 36 above, perform the operation C + C'. Do you get a symmetric matrix?

>> C + C'

ans =

2 4 -2 2
4 10 5 0
-2 5 14 -3
2 0 -3 6

We get a symmetric matrix.

In Exercise 36 above, extract the diagonal of the matrix C.

>> diag(C)

ans =

1
5
7
3

In Exercise 36 above, extract the upper triangular part and the lower triangular part of the matrix C.

>> triu(C)

ans =

1 2 -3 0
0 5 2 -3
0 0 7 -2
0 0 0 3

>> tril(C)

ans =

1 0 0 0
2 5 0 0
1 3 7 0
2 3 -1 3

In Exercise 36 above, determine the determinant and trace of the matrix C. Do you get scalars?

>> det(C)

ans =

-13

>> trace(C)

ans =

16

Both the determinant and trace are scalars (numbers in this example).

In Exercise 36 above, determine the inverse of the matrix C.

>> inv(C)

ans =

-10.5385 6.3846 -6.0000 2.3846
5.3077 -3.0769 3.0000 -1.0769
-0.3077 0.0769 -0.0000 0.0769
1.6154 -1.1538 1.0000 -0.1538

In Exercise 41 above, multiply the matrix C by its inverse matrix. Do you get the identity matrix?

>> C*inv(C)

ans =

1.0000 0.0000 0.0000 0.0000
-0.0000 1.0000 0 0.0000
0 0.0000 1.0000 0
0.0000 0.0000 0 1.0000

We get the identity matrix.

In Exercise 36 above, determine the norm of the matrix C.

>> norm(C)

ans =

9.5800

In Exercise 36 above, determine the eigenvalues of the matrix C.

>> eig(C)

ans =

-0.0702
3.9379 + 2.6615i
3.9379 - 2.6615i
8.1944

We get four eigenvalues – some of them are complex numbers.

In Exercise 36 above, determine the coefficients of the characteristic polynomial of the matrix C.

>> poly(C)

ans =

1.0000 -16.0000 86.0000 -179.0000
-13.0000
We get a fourth-degree polynomial with five coefficients.

In Exercise 36 above, determine the rank of the matrix C.

>> rank(C)

ans =

4

Generate a square random matrix of size 5.

>> rand(5)

ans =

0.9501 0.7621 0.6154 0.4057 0.0579
0.2311 0.4565 0.7919 0.9355 0.3529
0.6068 0.0185 0.9218 0.9169 0.8132
0.4860 0.8214 0.7382 0.4103 0.0099
0.8913 0.4447 0.1763 0.8936 0.1389

Generate a square magic matrix of size 7. What is the sum of each row, column or diagonal in this matrix.

>> magic(7)

ans =

30 39 48 1 10 19 28
38 47 7 9 18 27 29
46 6 8 17 26 35 37
5 14 16 25 34 36 45
13 15 24 33 42 44 4
21 23 32 41 43 3 12
22 31 40 49 2 11 20
The sum of each row, column or diagonal is 175.

Generate the following two symbolic matrices:

>> syms x

>> X = [1 x ; x-2 x^2]

X =

[ 1, x]
[ x-2, x^2]

>> Y = [x/2 3*x ; 1-x 4]

Y =

[ 1/2*x, 3*x]
[ 1-x, 4]

In Exercise 49 above, perform the matrix subtraction operation X – Y to obtain the new matrix Z.

>> Z = X-Y

Z =

[ 1-1/2*x, -2*x]
[ 2*x-3, x^2-4]

In Exercise 50 above, determine the transpose of the matrix Z.

>> Z'

ans =

[ 1-1/2*conj(x), -3+2*conj(x)]
[ -2*conj(x), -4+conj(x)^2]
In the above, it is assumed that x is a complex variable.

In Exercise 50 above, determine the trace of the matrix Z.

>> trace(Z)

ans =

-3-1/2*x+x^2

In Exercise 50 above, determine the determinant of the matrix Z.

>> det(Z)

ans =

5*x^2-4-1/2*x^3-4*x

In Exercise 50 above, determine the inverse of the matrix Z.

>> inv(Z)

ans =

[ -2*(x^2-4)/(-10*x^2+8+x^3+8*x),
-4*x/(-10*x^2+8+x^3+8*x)]
[ 2*(2*x-3)/(-10*x^2+8+x^3+8*x),
(x-2)/(-10*x^2+8+x^3+8*x)]

The inverse of Z obtained above may be simplified by factoring out the determinant that appears common in the denominators of all the elements of the matrix.

8. Programming

Write a script of four lines as follows: the first line should be a comment line, the second and third lines should have the assignments cost = 200 and sale_price = 250, respectively. The fourth line should have the calculation profit = sale_price – cost. Store the script in a script file called example8.m. Finally run the script file.

% This is an example
cost = 200
sale_price = 250
profit = sale_price – cost

Now, run the above script as follows:

>> example8

cost =

200

sale_price =

250

profit =

50

Write a function of three lines to calculate the volume of a sphere of radius r. The first line should include the name of the function which is volume(r). The second line should be a comment line. The third line should include the calculation of the volume of the sphere which is . Store the function in a function file called volume.m then run the function with the value of r equal to 2 (no units are used in this exercise).

function volume(r)
% This is a function
volume = (4/3)*pi*r^3

Now, run the above function as follows:

>> volume(2)

volume =

33.5103

Write a function with two arguments to calculate the area of a rectangle with sides a and b. The function should have three lines. The first line should include the name of the function which is RectangleArea(a,b). The second line should be a comment line. The third line should include the calculation of the area of the rectangle with is the product a*b. Store the function in a function file called RectangleArea.m then run the function twice as follow: the first execution with the values 3 and 6, while the second execution with the values 2.5 and 5.5.

function RectangleArea(a,b)
% This is another example of a function
RectangleArea = a*b

Now, run the above function as follows:

>> RectangleArea(3,6)

RectangleArea =

18

>> RectangleArea(2.5,5.5)

RectangleArea =

13.7500

Write a script containing a For loop to compute the vector x to have the values x(n) = n³ where n has the range from 1 to 7. Include a comment line at the beginning. Store the script in a script file called example9.m then run the script and display the values of the elements of the vector x.

% This is an example of a FOR loop
for n = 1:7
x(n) = n^3;
end

Now, run the above script as follows:

>> example9
>> x

x =

1 8 27 64 125 216 343

Write a script containing two nested For loops to compute the matrix y to have the values y(m,n) = m²-n² where both m and n each has the range from 1 to 4. Include a comment line at the beginning. Store the script in a script file called example10.m then run the script and display the values of the elements of the matrix y.

%This is an example of another FOR loop
for n = 1:4
for m = 1:4
y(m,n) = m^2 - n^2;
end
end

Now, run the above script as follows:

>> example10
>> y

y =

0 -3 -8 -15
3 0 -5 -12
8 5 0 -7
15 12 7 0

Write a script containing a While loop using the two variables tol and n. Before entering the While loop, initialize the two variables using the assignments tol = 0.0 and n = 3. Then use the two computations n = n + 1 and tol = tol + 0.1 inside the While loop. Make the loop end when the value of tol becomes equal or larger than 1.5. Include a comment line at the beginning. Store the script in a script file called example11.m then run the script and display the values of the two variables tol and n.

% This is an example of a While loop
tol = 0.0;
n = 3;
while tol < 1.5
n = n + 1;
tol = tol + 0.1;
end

Now, run the above script as follows:

>> example11
>> n

n =

18

>> tol

tol =

1.5000

Write a function called price(items) containing an If construct as follows. Let the price of the items be determined by the computation price = items*130 unless the value of the variable items is greater than 5 – then in this case the computation price = items*160 should be used instead. Include a comment line at the beginning. Store the function in a function file called price.m then run the function twice with the values of 3 and 9 for the variable items. Make sure that the function displays the results for the variable price.

function price(items)
% This is an example of an If Elseif construct
price = items*130
if items > 5
price = items*160
end

Now, run the above function as follows:

>> price(3)

price =

390

>> price(9)

price =

1170

price =

1440

Write a function called price2(items) containing an If Elseif construct as follows. If the value of the variable items is less than 3, then compute the variable price2 by multiplying items by 130. In the second case, if the value of the variable items is less than 5, then compute the variable price2 by multiplying items by 160. In the last case, if the value of the variable items is larger than 5, then compute the variable price2 by multiplying the items by 200. Include a comment line at the beginning. Store the function in a function file called price2.m then run the function three times – with the values of 2, 4, and 6. Make sure that the function displays the results for the variable price2.

function price2(items)
% This is an example of an If Elseif construct
if items < 3
price2 = items*130
elseif items < 5
price2 = items*160
elseif items > 5
price2 = items*200
end

Now, run the above function as follows:

>> price2(2)

price2 =

260

>> price2(4)

price2 =

640

>> price2(6)

price2 =

1200

Write a function called price3(items) containing a Switch Case construct. The function should produce the same results obtained in Exercise 8 above. Include a comment line at the beginning. Store the function in a function file called price3.m then run the function three times – with the values of 2, 4, and 6. Make sure that the function displays the results for the variable price3.

function price3(items)
% This is an example of the Switch Case construct
switch items
case 2
price3 = items* 130
case 4
price3 = items* 160
case 6
price3 = items* 200
otherwise
price3 = 0
end

Now, run the above function as follows:

>> price3(2)

price3 =

260

>> price3(4)

price3 =

640

>> price3(6)

price3 =

1200

Write a script file to store the following symbolic matrix A then calculate its third power B = A³. Include a comment line at the beginning. Store the script in a script file called example12.m then run the script to display the two matrices A and B.

% This is an example with the Symbolic
Math Toolbox
syms x
A = [x/2 1-x ; x 3*x]
B = A^3

Now, run the above script as follows:

>> example12

A =

[1/2*x, 1-x]
[ x, 3*x]

B =

[ 1/2*x*(1/4*x^2+(1-x)*x)+7/2*(1-x)*x^2,
7/4*(1-x)*x^2+(1-x)*((1-x)*x+9*x^2)]
[ x*(1/4*x^2+(1-x)*x)+21/2*x^3,
7/2*(1-x)*x^2+3*x*((1-x)*x+9*x^2)]

Write a function called SquareRoot2(matrix) similar to the function SquareRoot(matrix) described at the end of this chapter but with the following change. Substitute the value of 1.5 instead of 1 for the symbolic variable x. Make sure that you include a comment line at the beginning. Store the function in a function file called SquareRoot2.m then run the function using the following symbolic matrix:

function SquareRoot2(matrix)
% This is an example of a function
with the Symbolic Math Toolbox
y = det(matrix)
z = subs(y,1.5)
if z < 1
M = 2*sqrt(matrix)
else
M = sqrt(matrix)
end

Now, run the above function as follows:

>> syms x
>> M = [2 x 0 ; 3-x 5 -x ; x+2 1 3]

M =

[ 2, x, 0]
[ 3-x, 5, -x]
[ x+2, 1, 3]

>> SquareRoot2(M)

y =

30-7*x+x^2-x^3

z =

18.3750

M =

[ 2^(1/2), x^(1/2), 0]
[(3-x)^(1/2), 5^(1/2), (-x)^(1/2)]
[(x+2)^(1/2), 1, 3^(1/2)]

9. Graphs

Solve all the exercises using MATLAB. All the needed MATLAB commands for these exercises were presented in this chapter.

Plot a two-dimensional graph of the two vectors x = [1 2 3 4 5 6 7] and y = [10 15 23 43 30 10 12] using the plot command.

>> x = [1 2 3 4 5 6 7]

x =

1 2 3 4 5 6 7

>> y = [10 15 23 43 30 10 12]
y =

10 15 23 43 30 10 12

>> plot(x,y)

In Exercise 1 above, add to the graph a suitable title along with labels for the x-axis and the y-axis.

>> hold on;
>> title('This is an exercise')
>> xlabel('x-axis')
>> ylabel('y-axis')

Plot the mathematical function y = 2x³ + 5 in the range x from -6 to +6. Include a title for the graph as well as labels for the two axes.

>> x = [-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6];
>> y = 2*x.^3 +5;
>> plot(x,y)
>> hold on;
>> title('This is another exercise')
>> xlabel('values of x')
>> ylabel('values of y')

Repeat the plot of Exercise 3 above but show the curve with a blue dashed line with circle symbols at the plotted points.

plot(x,y,'b--o')

Plot the two mathematical functions and such that the two curves appear on the same 3 diagram. Make the range for x between 0 and 3π with increments of Distinguish the two curves by plotting one with a dashed line and the other one with a dotted line. Include the title and axis information on the graph as well as a grid and a legend.

>> x = 0:pi/4:3*pi;
>> y = 2*sin(x/3);
>> z = 2*cos(x/3);
>> plot(x,y,'--',x,z,':')
>> hold on;
>> grid on;
>> title('This is an exercise in plotting two curves on the same diagram')
>> xlabel('x')
>> ylabel('2sin(x/3) and 2cos(x/3)')
>> legend('2sin(x/3)','2cos(x/3)')

Plot the following four mathematical functions each with its own diagram using the subplot command. The functions are and v = x² + 3. Use the vector x = [1 2 3 4 5 6 7 8 9 10] as the range for x. No need to show title or axis information.

>> x = [1 2 3 4 5 6 7 8 9 10];
>> y = 2*x.^3 -4;
>> z = x+1;
>> w = 2 - sqrt(x);
>> v = x.^2 +3;
>> subplot(2,2,1);
>> hold on;
>> plot(x,y);
>> subplot(2,2,2);
>> plot(x,z);
>> subplot(2,2,3);
>> plot(x,w);
>> subplot(2,2,4);
>> plot(x,v);

Use the plot3 command to show a three-dimensional curve of the equation z = 2sin(xy). Use the vector x = [1 2 3 4 5 6 7 8 9 10] for both x and y. No need to show title or axis information.

>> x = [1 2 3 4 5 6 7 8 9 10];
>> y = [1 2 3 4 5 6 7 8 9 10];
>> z = 2*sin(x.*y);
>> plot3(x,y,z)

Use the mesh command to plot a three-dimensional mesh surface of elements of the matrix M given below. No need to show title or axis information.

>> M = [0.1 0.2 0.5 0.7 0.3 ; 0.2 0.8 0.5 0.6 0.3 ; 0.5 0.9 0.9 0.4 0.4 ; 0.4 0.4 0.5 0.7 0.9 ; 0.1 0.3 0.4 0.6 0.8]

M =

0.1000 0.2000 0.5000 0.7000 0.3000
0.2000 0.8000 0.5000 0.6000 0.3000
0.5000 0.9000 0.9000 0.4000 0.4000
0.4000 0.4000 0.5000 0.7000 0.9000
0.1000 0.3000 0.4000 0.6000 0.8000

>> mesh(M)

Use the surf command to plot a three-dimensional surface of the elements of the matrix M given in Exercise 8 above. No need to show title or axis information.

>> surf(M)

Use the contour command to plot a contour map of the elements of the matrix M given in Exercise 8 above. No need to show the contour values on the graph.

>> contour(M)

Use the surfc command to plot a three-dimensional surface along with contours underneath it of the matrix M given in Exercise 8 above. No need to show title or axis information.

>> surfc(M)

10. Solving Equations

Solve the following linear algebraic equation for the variable x. Use the roots command.

3x + 5 = 0

>> p = [3 5]

p =

3 5

>> roots(p)

ans =

-1.6667

Solve the following quadratic algebraic equation for the variable x. Use the roots command.

x² + x +1 = 0

>> p = [1 1 1]

p =

1 1 1

>> roots(p)

ans =

-0.5000 + 0.8660i
-0.5000 - 0.8660i

Solve the following algebraic equation for x. Use the roots command.

3x⁴ - 2x³ + x - 3 = 0

>> p = [3 -2 0 1 -3]

p =

3 -2 0 1 -3

>> roots(p)

ans =

1.1170
0.2426 + 0.9477i
0.2426 - 0.9477i
-0.9355

Solve the following system of linear simultaneous algebraic equations for the variables x and y. Use the inverse matrix method.

3x + 5y = 9
4x - 7y = 13

>> A = [3 5 ; 4 -7]

A =

3 5
4 -7

>> b = [9 ; 13]

b =

9
13

>> x = inv(A)*b

x =

3.1220
-0.0732

In Exercise 4 above, solve the same linear system again but using the Gaussian elimination method with the backslash operator.

>> A = [3 5 ; 4 -7]

A =

3 5
4 -7

>> b = [9 ; 13]

b =

9
13

>> x = A\b

x =

3.1220
-0.0732

Solve the following system of linear simultaneous algebraic equations for the variables x, y, and z. Use Gaussian elimination with the backslash operator.

2x - y + 3z = 5
4x + 5z = 12
x + y + 2z = -3

>> A = [2 -1 3 ; 4 0 5 ; 1 1 2]
A =

2 -1 3
4 0 5
1 1 2

>> b = [5 ; 12 ; -3]

b =

5
12
-3

>> x = A\b

x =

10.0000
-1.8000
> -5.6000

Solve the following linear algebraic equation for the variable x in terms of the constant a.

2x + a = 5

>> syms x
>> syms a
>> solve('2*x + a - 5 = 0')

ans =

-1/2*a+5/2

Solve the following quadratic algebraic equation for the variable x in terms of the constants a and b.

x² + ax + b = 0

>> syms x
>> syms a
>> syms b
>> solve('x^2 + a*x + b = 0')

ans =

-1/2*a+1/2*(a^2-4*b)^(1/2)
-1/2*a-1/2*(a^2-4*b)^(1/2)

Solve the following nonlinear equation for the variable x.

2e^x+ 3cos x = 0

>> syms x

>> solve('2*exp(x) + 3*cos(x) = 0')

ans =

-1.6936679094546100269188394124169

The above answer is approximated as -1.694.

Solve the following system of linear simultaneous algebraic equations for the variables x and y in terms of the constant c.

2x − 3cy =5
cx + 2y =7

>> syms x
>> syms y
>> syms c
>> [x,y] = solve('2*x -3*c*y -5 = 0', 'c*x + 2*y - 7 = 0')

x =

(21*c+10)/(4+3*c^2)

y =

-1/(4+3*c^2)*(-14+5*c)

Solve the following system of nonlinear simultaneous algebraic equations for the variables x and y.

3 x² − 2+ + y = 7
xy + x = 5

>> syms x
>> syms y
>> [x,y] = solve('3*x^2 - 2*x + y - 7 = 0', 'x*y + x - 5 = 0')

x

= 5/3
1/2*5^(1/2)-1/2
-1/2-1/2*5^(1/2)

y =

2
3/2+5/2*5^(1/2)
3/2-5/2*5^(1/2)