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)