Use MATLAB to compute the determinants of the matrices A+B, A-B, AB, A-1, and BT. (Recall that in MATLAB, BT is written as B'.) Which of the above matrices are not invertible? Explain your reasoning. Suppose that you didn't know the entries of A and B, but you did know their determinants. Which of the above determinants would you still be able to compute from this information, even without having A or B at hand? Explain your reasoning.

The problem is solved in Matlab and the code with explanation is provided below.

Explanation:

(a) Use MATLAB to compute the determinants of the matrices A+B, A-B, AB, A⁻¹, and BT

We can use the Matlab function det() to calculate the determinants and inv() function is used to calculate the inverse.

A = [8 11 2 8; 0 -7 2 -1; -3 -7 2 1; 1 1 2 4]

B = [1 -2 0 5; 0 7 1 5; 0 4 4 0; 0 0 0 2]

a=det(A+B)

b=det(A-B)

c=det(A*B)

d=det(inv(A))

e=det(B')

Output:

A =

    8    11     2     8

    0    -7     2    -1

   -3    -7     2     1

    1     1     2     4

B =

    1    -2     0     5

    0     7     1     5

    0     4     4     0

    0     0     0     2

a =    4.4964e-15 (this is basically zero)

b =    1.0380e+03

c =     3648

d =    0.0132

e =    48

(b) Which of the above matrices are not invertible? Explain your reasoning

As you can see in part (a) the determinant of A+B is approximately zero  and we know that the determinant of an invertible matrix can never be zero hence A+B must  be non-invertible.

(c) Suppose that you didn't know the entries of A and B, but you did know their determinants. Which of the above determinants would you still be able to compute from this information, even without having A or B at hand? Explain your reasoning.

If we did not had original matrix A and B but we had det(A) and det(B) then we could have found  the following by using the properties of determinants:

1. A*B because det(A)*det(B) = det(A*B)

2. inv(A) because inv(A) = 1/det(A)

3. B' because det(B') = det(B)

a  social news website  is an  internet  website  that features  user-posted stories. such stories are ranked based on popularity, as voted on by other users of the site or by website administrators. users typically comment online on the news posts and these comments may also be ranked in popularity. since their emergence with the birth of  web 2.0, social news sites have been used to link many types of information, including news, humor, support, and discussion. all such websites allow the users to submit content and each site differs in how the content is moderated. on the  slashdot  and  fark  websites,  administrators  decide which articles are selected for the front page. on  reddit  and  digg, the articles that get the most votes from the community of users will make it to the front page. many social news websites also feature an online comment system, where users discuss the issues raised in an article. some of these sites have also applied their voting system to the comments, so that the most popular comments are displayed first. some social news websites also have a  social networking  function, in that users can set up a user profile and follow other users' online activity on the website.