Question

Recall that if A is an m × n matrix and B is a p × q matrix, then the product C = AB is defined if and only if n = p, in which case C is an m × q matrix. (a) Write a function M-file that takes as input two matrices A and B, and as output produces the product by columns of the two matrix. For instance, if A is 3 × 4 and B is 4 × 5, the product is given by the matrix C = [A*B(:,1), A*B(:,2), A*B(:,3), A*B(:,4), A*B(:,5)] The function file should work for any dimension of A and B and it should perform a check to see if the dimensions match (Hint: use a for loop to define each column of C). Call the file columnproduct.m. Test your function on a random 3×5 matrix A and a random 5×2 matrix B . Compare the output with A*B. Repeat with 4 × 6 and 6 × 2 matrices and with 4 × 6 and 2 × 6 matrices. Use the command rand to generate the random matrices for testing. Include in your lab report the function M-file and the output obtained by running it. (b) Write a function M-file that takes as input two matrices A and B, and as output produces the product by rows of the two matrices. For instance, if A is 3 × 4 and B is 4 × 5, the product AB is given by the matrix C = [A(1,:)*B; A(2,:)*B; A(3,:)*B] The function file should work for any dimension of A and B and it should perform a check to see if the dimensions match (Hint: use a for loop to define each row of C). Call the file rowproduct.m. Test your function on a random 3×5 matrix A and a random 5×2 matrix B . Compare the output with A*B. Repeat with 4 × 6 and 6 × 2 matrices and with 4 × 6 and 2 × 6 matrices. Use the command rand to generate the random matrices for testing. Include in your lab report the function M-file and the output obtained by running it.

Answer #1

n×n-matrix M is symmetric if M = M^t. Matrix M is
anti-symmetric if M^t = -M.
1. Show that the diagonal of an anti-symmetric matrix are
zero
2. suppose that A,B are symmetric n × n-matrices. Prove that AB
is symmetric if AB = BA.
3. Let A be any n×n-matrix. Prove that A+A^t is symmetric and A
- A^t antisymmetric.
4. Prove that every n × n-matrix can be written as the sum of a
symmetric and anti-symmetric matrix.

Creating a 6x8 matrix with rank 2 In general, rank(AB) <=
min(rank(A), rank(B)). If A and B are random matrices, the relation
should be an equality.
Try to create a 6x8 matrix with rank 2 by modifying the above
procedure. (hint: mx2 times 2xn is mxn). Verify the matrix is rank
2 with the rank command

2. Write the output matrix “v”
t = [2:4];
k = [1:3];
v = t.*k – k.^2
4. Given: D = [1 2 3 4 5 6 7 8 9] (3x3) . Which command will
extract the submatrix [1 2 3 4 5 6] (2x3) ?
a. D[1:2,1:3]
b. D(1,2 ;1,3)
c. [D(1:2),D(1:3)]
d. D(1:2,1:3)
14. What will be the dimension of matrix B?
B=[ones(3) zeros(3) rand(3); 2*eye(9)]
18. Find the value of “C”
A=1:2:10;
B=linspace(1,5,5);
C = length(A)*B(2)+A(5)*B(3);
19....

8. Consider a 4 × 2 matrix A and a 2 × 5 matrix B .
(a) What are the possible dimensions of the null space of AB?
Justify your answer.
(b) What are the possible dimensions of the range of AB Justify
your answer.
(c) Can the linear transformation define by A be one to one?
Justify your answer.
(d) Can the linear transformation define by B be onto? Justify
your answer.

By Using Matlab
Matcal (A, B) returns matrix A and B products, while printing
the matrix computation formula in outmat. txt file as follows. (The
sizes of A and B are very variable. In other words, it shall be
possible to perform operation for matrix operation of 2 x 2 size,
and it shall also be possible to calculate and output for 100 X 100
matrix models. )
>> A = [1 23;4 56 ; 79]
>> B = [0...

a).For the reduction of matrix determine the elementary matrices
corresponding to each operation. M= 1 0 2 1 5
1 1 5 2 7
1 2 8 4 12 b). Calculate the product P of these elementary
matrices and verify that PM is the end result.

Perform the following matrix multiplications
A = [2 5 6]
[-3 0 1]
B = [ 1]
[ 3]
[-4]
C = [-2 10 1]
6a. Find AB
6b. Find BC
6c. Find AC

1- Create a new script that is called HW9.m and save it. The
script will do the following. Test your code for each of the cases
and upload your script as a .m file. [15 points]
a. Create a random integer numbers that goes from 0 to 5 and
assign it to a variable y using rand and round commands. Hint: Note
that rand function only creates random number between 0 and 1. You
need to scale the values to...

MATLAB:
Write a function called matrix_problem1 that
takes a matrix A of positive integers as its sole input. If the
assumption is wrong, the function returns an empty matrix.
Otherwise, the function doubles every odd element of A and returns
the resulting matrix. Notice that the output matrix will have all
even elements. For example, the call B = matrix_problem([1 4; 5 2;
3 1], will make B equal to [2 4; 10 2; 6 2]. The function should
work...

If the matrix A is 4×2, B is 3×4, C is 2×4, D is 4×3,
and E is 2×5, what are the
dimensions of the following expressions
a) A^TD + CB^T
b) (B + D^T ) A
c) CA + CB^T

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