Create the following three matrices A 1 3 5 2 2 4 2 0 6 B

Create the following three matrices: A = [1 -3 5 2 2 4 -2 0 6] B = [0 -2 1 5 1 -6 2 7 -1] C = [-3 4 1 0 8 2 -3 5 3] (a) Calculate A + B and B + A to show that addition of matrices is commutative. (b) Calculate A + (B + C) and (A + B) + C to show that addition of matrices is associative. (c) Calculate 3 (A + C) and 3A +3C to show that, when matrices are multiplied by a scalar, the multiplication is distributive. (d) Calculate A*(B + C) and A*B + A*C to show that matrix multiplication is distributive. Use the matrices A, B, and C from the previous problem to answer the following: (a) Does A*B = B*A ? (b) Does A*(B*C) = (A*B)*C? (c) Does (A*B)^t = A^t*B^t (^t means transpose) (d) Does (A + B)^t = A^t + B^t?

Solution

(a)
%Matrix A of size 3 x3 %
A=[1 -3 5; 2 2 4; -2 0 6];
%Matrix B of size 3 x3 %
B=[0 -2 1; 5 1 -6; -2 0 6];
%Matrix C of size 3 x3 %
C=[-3 4 -1; 0 8 2; -3 5 3];

disp(\'Checking A+B = B+A\')
disp(\'A+B\')
A+B
disp(\'B+A\')
B+A

Result fro matlab console:
Checking A+B = B+A
A+B
ans =
1 -5 6
7 3 -2
-4 0 12
B+A
ans =
1 -5 6
7 3 -2
-4 0 12

A+B and B+A is commutative

(b)

%Matrix A of size 3 x3 %
A=[1 -3 5; 2 2 4; -2 0 6];
%Matrix B of size 3 x3 %
B=[0 -2 1; 5 1 -6; -2 0 6];
%Matrix C of size 3 x3 %
C=[-3 4 -1; 0 8 2; -3 5 3];

disp(\'Checking A+(B+C) = (A+B)+C)\')
disp(\'A+(B+C)\')
(A+B)+C
disp(\'(A+B)+C\')
A+(B+C)

Result:
Checking A+(B+C) = (A+B)+C)
A+(B+C)
ans =
-2 -1 5
7 11 0
-7 5 15
(A+B)+C
ans =
-2 -1 5
7 11 0
-7 5 15

(c)
%Matrix A of size 3 x3 %
A=[1 -3 5; 2 2 4; -2 0 6];
%Matrix B of size 3 x3 %
B=[0 -2 1; 5 1 -6; -2 0 6];
%Matrix C of size 3 x3 %
C=[-3 4 -1; 0 8 2; -3 5 3];

disp(\'Checking 3*(A+C) = 3*A+3*C\')
disp(\'3*(A+B)\')
3*(A+B)
disp(\'3*A+3*B\')
3*A+3*B

Result:
Checking 3*(A+C) = 3*A+3*C
3*(A+B)
ans =
   3 -15 18
21   9 -6
-12   0 36
3*A+3*B
ans =
   3 -15 18
21   9 -6
-12   0 36

(d)

%Matrix A of size 3 x3 %
A=[1 -3 5; 2 2 4; -2 0 6];
%Matrix B of size 3 x3 %
B=[0 -2 1; 5 1 -6; -2 0 6];
%Matrix C of size 3 x3 %
C=[-3 4 -1; 0 8 2; -3 5 3];

disp(\'Checking A*(B+C) = A*B+A*C\')
disp(\'A*(B+C)\')
A*(B+C)
disp(\'A*B+A*C\')
A*B+A*C

Result:
Checking A*(B+C) = A*B+A*C
A*(B+C)
ans =
-43   0 57
-16 42 28
-24 26 54
A*B+A*C
ans =
-43   0 57
-16 42 28
-24 26 54

(30)
(a)
%Matrix A of size 3 x3 %
A=[1 -3 5; 2 2 4; -2 0 6];
%Matrix B of size 3 x3 %
B=[0 -2 1; 5 1 -6; -2 0 6];
%Matrix C of size 3 x3 %
C=[-3 4 -1; 0 8 2; -3 5 3];

disp(\'Checking A*B = B*A\')
disp(\'A*B\')
A*B
disp(\'B*A\')
B*A

Result:
Checking A*B = B*A
A*B
ans =
-25 -5 49
   2 -2 14
-12   4 34
B*A
ans =
-6 -4 -2
19 -13 -7
-14   6 26

AB is not equal to BC
(b)
%Matrix A of size 3 x3 %
A=[1 -3 5; 2 2 4; -2 0 6];
%Matrix B of size 3 x3 %
B=[0 -2 1; 5 1 -6; -2 0 6];
%Matrix C of size 3 x3 %
C=[-3 4 -1; 0 8 2; -3 5 3];

disp(\'Checking A*(B*C) = (A*B)*C)\')
disp(\'A*(B*C)\')
A*(B*C)
disp(\'(A*B)*C\')
(A*B)*C

Result:
Checking A*(B*C) = (A*B)*C)
A*(B*C)
ans =
-72 105 162
-48 62 36
-66 154 122
(A*B)*C
ans =
-72 105 162
-48 62 36
-66 154 122

(c)
%Matrix A of size 3 x3 %
A=[1 -3 5; 2 2 4; -2 0 6];
%Matrix B of size 3 x3 %
B=[0 -2 1; 5 1 -6; -2 0 6];
%Matrix C of size 3 x3 %
C=[-3 4 -1; 0 8 2; -3 5 3];

disp(\'Checking transpose of (A*B*) = transpose of A*tranpose of B\')

%% tranpose of a matrix is .\' for A*B
(A*B).\'
%% tranpose of a matrix is .\' for B*A
(B*A).\'

Result:
Checking transpose of (A*B*) = transpose of A*tranpose of B
ans =
-25   2 -12
-5 -2   4
49 14 34
ans =
-6 19 -14
-4 -13   6
-2 -7 26

(d)
%Matrix A of size 3 x3 %
A=[1 -3 5; 2 2 4; -2 0 6];
%Matrix B of size 3 x3 %
B=[0 -2 1; 5 1 -6; -2 0 6];
%Matrix C of size 3 x3 %
C=[-3 4 -1; 0 8 2; -3 5 3];

disp(\'Checking transpose of (A+B) = transpose of A* transpose pf B\')
(A+B).\'
%% tranpose of a matrix is .\' for B*A
(B+A).\'

Result :
Checking transpose of (A+B) = transpose of A* transpose pf B
ans =
1 7 -4
-5 3 0
6 -2 12
ans =
1 7 -4
-5 3 0
6 -2 12

Both are equal so transpose of (A+B)= transpose of A + tranpose of B