Hi, I need help with this. Write up your solution and results so that it could be understood by someone else completing the unit, including all required working, investigations and relevant observations (VERY IMPORTANT).

PLEASE INCLUDE DETAILED EXPLANATION AND WORKING

ALSO DONT COPY FROM OTHER QUESTIONS SINCE THEY ARE ALL WRONG

(A)

c d , algebraically determine conditions that would lead to (a) distinct eigenvalues, (b) a repeated eigenvalue with only one non-linearly independent vector, and (c) a i. For a 2 X 2 matrix [a b] repeated eigenvalue with two linearly independent vectors. ii. Hence provide original examples of each type. iii. Can you nd any similar conditions for the different cases for a 3 X 3 matrix? Consider 4 column vectors: , with r, y, z ER. i. Identify values of x, y, z such that the four vectors are linearly independent. Be sure to provide reasoning (e.g., along the lines of the statement in 4.1.4 of the Study Guide) and demonstrate that linear independence is satisfied. ii. Identify values of x, y, z such that the four vectors are linearly dependent, but any three of the vectors taken together are linearly independent. iii. In part (ii), showing the rank of the 4 x 4 matrix formed from your vectors to be 3 would not generally be a sufficient argument. Why not