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Problems Problem 1.1 :

Matrix generation

i) Create the following matrixes

ii) Create C : a 3×3 identity matrix. Use the most convenient method.

Problem 1.2 : Basic matrix operations Use the matrixes you made in Problem 1.1. i) Compare the results of A* B and A.* B . Explain what MATLAB did. ii)Does A* B  B * A ? Explain. iii)Does A.* B  B.* A ? Explain. iv)Create matrix D  B1 . What is B * D ? v)Compare A / B and A \ B . Can you explain what is going on? vi)Create a new 3×3 matrix E with all diagonal elements equal to 5 and other elements equal to 0.

Problem 1.3 : Understanding data types Use the matrixes you made in Problem 1.1. vii)What are the data types of the elements of matrixes A , B , and C ? viii)How many bits of memory are required to store the information in matrix matlabassignmentexperts.com A?

iii)Convert A and B to 8 bit signed integer matrixes and define them as matrix F and G respectively (check out function int8( ) in the help menu). iv)Calculate F  G and F *G .Explain results.

Problem 1.4 : Submatrix extraction and plotting You are given the trajectory of a ball in a MATLAB file: ball.mat. (Please, download it from MIT Server site.) The first column contains 100 time points and the second column contains the corresponding 100 positions of the ball. The third and the fourth column contain a second trial of the same experiment. v)Extract the time data from column 1 and assign that to a new matrix “t”. vi)Extract the position data from column 2 and assign that to a new matrix “x”. vii)Plot “x” vs. “t”. viii)Extract only the first 50 time points from column 3 and make a new matrix “t2”. ix)Extract only the first 50 positions from column 4 and make a new matrix “x2”. x)Plot “x2” vs. “t2”. xi)Superimpose both trajectories. Plot them in different color. matlabassignmentexperts.com

Solutions Problem 1.1 :

Matrix generation

i) You can make one of following methods listed below: A = [ [1 2 3]' [4 5 6]' [7 8 9]']; A(1,:) = [1 4 7]; A(2,:) = [2 5 8]; A(3,:)=[3 6 9]; A(:,1) = [1 2 3]; A(:,2) = [4 5 6]; A(:,3)=[7 8 - A(1,1) 9]; = 1; A(2,1) = 2; A(3,1) = 3; A(1,2) = 4; A(2,2) = 5; A(3,3) = 9;

A(3,2) = 6;

A(1,3) = 7; A(2,3) =

8;

and so on…

B = [1 2 3; 0 3 1; 2 1 1]; or equivalent ways shown in creating matrix A . >> A = [1 4 7; 2 5 8; 3 6 9] A=

>> B = B=

1 2 3

4 5 6 [1 2

1 0 2

2 3 1

7 8 9 3; 0 3

1; 2 1 1]

3 1 1

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ii) The most convenient method is to use eye() function (Please, see eye() in help menu.) >> C = eye(3) % eye(N) generate identity matrix with N x N

C = 1 0 0

0 1 0

0 0 1

Problem 1.2 : Basic matrix operations i) A*B gives matrix multiplication operation of matrices A and B (inner product between corresponding row of matrix A and corresponding column of matrix B ), whereas A.*B provides element by element multiplication only between corresponding elements of matrices A and B .

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>> A*B ans = 15 18 21 >> A.*B ans = 1 0 6

% Matrix multiplication 21 27 33

14 19 24 % Element by element multiplication

8 15 6

21 8 9

ii) In matrix multiplication, commutativity does not generally hold. >> A*B ans = 15 18 21 >> B*A ans = 14

9 7

21 27 33

14 19 24 % Result is not same as A*B

32 21 19

50 33 31

iii) Following two operations are same since . * holds commutativity.

>> A.*B ans =

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v) A/B is roughly same to A*inv(B) (solution to equation XB  A ) if B is a square matrix, and A\B is roughly same to inv(A)*B (solution to equation AX  B ) if A is a square matrix. (For more information see mldivide() and mrdivide() in help menu.) Since matrix A is singular, inverse of A does not exists. (determinant of A is equal to 0. Check it with det(A). ) Therefore, MATLAB gives the below warning to us for A\B.

>> A/B ans =

% equal to A*inv(B) 2.6667 2.8333 3.0000

-0.1667 -0.8333 -0.0833 -0.4167 0 0 >> A\B % equal to inv(A)*B Warning: Matrix is singular to working precision.

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1 0 6 >> B.*A ans = 1 0 6

8 15 6 8 15 6

21 8 9 % Result is same as A.*B 21 8 9

iv) B*D is a 3×3 identity matrix, since B * D  B * B1  I . >> D=B^-1 % or D= -0.1667 -0.0833 -0.1667 0.4167 0.5000 -0.2500 >> B*D % Generates 3x3 identity matrix ans = 0 0 1 1 1 0 1 0 1

>> A/B ans =

D=inv(B) 0.5833 0.0833 -0.2500

% equal to A*inv(B) 2.6667 2.8333 3.0000

-0.1667 -0.8333 -0.0833 -0.4167 0 0 >> A\B % equal to inv(A)*B Warning: Matrix is singular to working precision.

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ans =

NaN Inf -Inf

NaN -Inf Inf

NaN Inf -Inf

vi) aij  5ij i, j  1, 2, 3

where aij is i-th row and j-th column element of matrix.

(diagonal matrix with all 5’s)

>> E=5*eye(3) % or E=diag([5 5 5]), and so on 5 0 0 E =

0 0

5 0

0 5

vii) Understanding data types i) By default, MATLAB assigns variables to ‘double’ data type variables.

Therefore, A , B , and C are matrices whose elements are all ‘double’ data types. You can check it with ‘whos’ command in MATLAB. >> whos Name A B C

Size 3x3 3x3 3x3

Bytes 72 72 72

Class double double double

Attributes

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ii)‘double’ data type variable requires 8 bytes to store a variable value, and each element in matrix A is 8 byte long. Since matrix A has 3×3=9 elements (or variables), matrix A needs 72 bytes. Therefore, 576 bits are required to store the information in matrix A, since 1 byte consists of 8 bits. iii)New matrices F and G are displayed same as matrices A and B respectively, but they are stored in the computer with different data format. >> F=int8(A) F = 1 4 7 2 5 8 3 6 G= 1 2 3 9 0 3 1 >> 2 1 1 G=int8(B ) Addition between only same integer data types is supported in MATLAB, but iv) matrix multiplication between any integer data types is not. Therefore, F+G produces matrix addition and F*G generates error as below. (See ‘Arithmatic operators’ in help menu.) matlabassignmentexperts.com

>> F+G ans = 2 2 5 >> F*G ??? Erro Integer At least

% Only allowed if both are same integer types 6 8 7

10 9 10 % Never allowed if at least one is integer data type r using ==> mtimes dat a types are not fully supported for this operation. e operand must be a scalar. on

viii) Submatrix extraction and plotting i) First, saved data should be retrieved with load(). ‘:’ can be used to select either whole row or whole column in the matrix. >> load ball.mat % load data from ball.mat >> t=A(:,1);% Extract 1st column data, and assign to ‘t’ ii)

>> x=A(:,2);% Extract 2nd column data, and assign to ‘x’

iii) >> plot(t,x);

% Plot t vs. x

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iv) 1:50 means selection from first element to 50th elements >> t2=A(1:50,3); %Extract first 50 elements from 3rd column of A v)

>> x2=A(1:50,4); %Extract first 50 elements from 4th column of A >> plot(t2,x2);% Plot t2 vs. x2

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vii) You can enumerate appropriate arguments for each graph (x data, y data, and 0

linespec) sequentially in the plot() function. ‘r’ means red color and ‘b’ indicates blue color. (See ‘linespec’ in help menu) Alternatively, you can also overlap several plots with ‘hold’ in help menu)

‘hold on’ and ‘hold off’ (See matlabassignmentexperts.com

>> plot(t,x,'r',t2,x2,'b'); or >>plot(t,x,'r'); hold on; plot(t2,x2,'b'); hold off; 7

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