PROBLEM 1 - Use MATLAB commands to efficiently (i.e. without keying each entry) enter the matrix:

Hint: Try help diag and represent S as the sum of two matrices having non-zero entries on different diagonals.

a) Compute S^k for k= 2, 3, 4.  Describe in words what happens when computing S^k.

b) By using this reasoning, what do you expect S¹¹ to be?

c) As part of a linear algebra quiz, your friend answers 'TRUE' to the question: Is it true or false that : If A*B=0 then one of the matrices A or B is the zero matrix? Based on your computations above, do you agree with him?

PROBLEM 2 - Suppose a linear transformation T has the property that T([1;3])=[5;4], and T([2;1])=[3;6] where [1;3] is, as in MATLAB, the column vector with entries 1 and 2. Let A denote the standard matrix of T.

a) The information above tells you that there are matrices U and V such as A ∗ U = V. Define U and V.  Hint: read the problem until the end.

b) Using inv(U), V and matrix multiplication, compute A.

c) Verify that you have the correct A by computing in MATLAB A ∗ [1;3] and A ∗ [2;1] and comparing with the values of T([1;3]) and T([2;1]), respectively.

d) Compute the expression det A · det U - det V. What general fact does this calculation illustrates?

e) Compute det (A + U) - (det A + det U). What general fact does this calculation illustrates?

PROBLEM 3 - Let A_?? be the ?? × ?? matrix with 1 on the main diagonal and 2 elsewhere.

a) For ?? = 4, 5, 6

1. Use Matlab pre-programmed matrices (eye, ones, zeros) and matrix operations, efficiently input A_??.

2. Compute A_??^-1 and display the result with rational entries.

b) Propose a general form for A_??^-1, expressed in terms on ??.

c) Check your theory for ?? = 6.

PROBLEM 4 - Consider the matrix A = [4, -2, 1, 5; 3, 8, 2, -1; 6, 8, 9, 2; 2, 3, -1, 0]. Compute the following five determinants and comment what general properties of determinants your computations at point's b-e illustrate:

(a) det(A); (b) det(A^T) where ^T stands for transposed; (c)det(A²); (d) det(2 A); (e ) det (A^-1).

PROBLEM 5: The color of light can be represented in a vector [R; G; B] where R= Amount of red; G= amount of green and B=amount of blue. The human eye and the brain transform the incoming signal into the signal [I; L; S] where I - intensity, L -long-wave signal and S - short wave signal and ?? = (??+??+??)/3; ?? = R - ??; S = B - (??+??)/2

a) Find the standard matrix P of the transformation from input [R; G; B] to output [I; L; S];

b) Consider a pair of yellow sunglasses for water sports that cuts out all blue light and passes all red and green light. Find the matrix A that represents the transformation incoming light undergoes as it passes through the sunglasses;

c) Find the matrix for the composed transformation that the light undergoes as it first passes through the sunglasses and then the eye;

d) Compute the change in the [I; L; S] output signal between the [I; L; S] output without and with the yellow sunglasses if the initial [R; G; B] input is [20; 35; 40].